[Chapter 1 - Problem 1]
Question (EN): Write a Python function to check if a given sequence (list of real numbers) is monotonic increasing or decreasing.
Question (JP): 与えられた数列（実数のリスト）が単調増加か単調減少かを判定するPython関数を書け。
Hint: Use all() with comparisons.
[Chapter 1 - Problem 2]
Question (EN): Implement a Python function to approximate the limit of a sequence a_n = (1 + 1/n)^n as n → ∞.
Question (JP): 数列 a_n = (1 + 1/n)^n の n→∞ の極限を近似するPython関数を実装せよ。
Hint: Use numpy for large n.
[Chapter 1 - Problem 3]
Question (EN): Simulate the convergence/divergence of the sequence a_n = (-1)^n using Python, and plot the first 50 terms.
Question (JP): 数列 a_n = (-1)^n の収束/発散をPythonでシミュレーションし、最初の50項をプロットせよ。
Hint: Use matplotlib.
[Chapter 1 - Problem 4]
Question (EN): Write a program to compute the decimal expansion of 1/7 up to 20 digits.
Question (JP): 1/7 の小数展開を20桁まで計算するプログラムを書け。
Hint: Use Python's decimal module.
[Chapter 1 - Problem 5]
Question (EN): Write a Python script to compute the limsup and liminf of the sequence a_n = sin(n).
Question (JP): 数列 a_n = sin(n) の上極限と下極限を求めるPythonスクリプトを書け。
Hint: Use numpy and slicing.
[Chapter 1 - Problem 6]
Question (EN): Generate 1000 random real numbers in [0,1] and compute their average. Does it converge to 0.5?
Question (JP): 区間[0,1]で1000個の乱数を生成し、その平均を計算せよ。0.5に収束するか確認せよ。
Hint: Use random module.
[Chapter 1 - Problem 7]
Question (EN): Write a Python program to check if a sequence is a Cauchy sequence.
Question (JP): 数列がコーシー列であるかを判定するPythonプログラムを書け。
Hint: Use epsilon definition and nested loops.
[Chapter 1 - Problem 8]
Question (EN): Implement a function that returns the n-th decimal digit of √2.
Question (JP): √2 の小数第 n 位の数字を返す関数を実装せよ。
Hint: Use decimal module for precision.
[Chapter 1 - Problem 9]
Question (EN): Write Python code to approximate e by summing 1/n! until convergence.
Question (JP): e を 1/n! の和で近似するPythonコードを書け。
Hint: Use math.factorial.
[Chapter 1 - Problem 10]
Question (EN): Write a program to compare the convergence speed of two sequences: (1+1/n)^n and n^(1/n).
Question (JP): 数列 (1+1/n)^n と n^(1/n) の収束速度を比較するプログラムを書け。
Hint: Use matplotlib for plotting.
[Chapter 1 - Problem 11]
Question (EN): Implement a Python script to test whether the harmonic series ∑ 1/n diverges by computing partial sums.
Question (JP): 調和級数 ∑ 1/n が発散することを部分和の計算で確認せよ。
Hint: Plot partial sums.
[Chapter 1 - Problem 12]
Question (EN): Write code to approximate the alternating harmonic series ∑ (-1)^(n+1)/n and check convergence.
Question (JP): 交代調和級数 ∑ (-1)^(n+1)/n の収束を確認するコードを書け。
Hint: Use itertools and numpy.
[Chapter 1 - Problem 13]
Question (EN): Implement a function that checks if a decimal expansion of a rational number is eventually periodic.
Question (JP): 有理数の小数展開が有限周期であるかを確認する関数を実装せよ。
Hint: Use fraction module.
[Chapter 1 - Problem 14]
Question (EN): Write Python code to compute the geometric mean of the first n natural numbers.
Question (JP): 最初の n 個の自然数の幾何平均を計算するPythonコードを書け。
Hint: Use math.prod.
[Chapter 1 - Problem 15]
Question (EN): Write a program that simulates the convergence of a_n = cos^n(x) for a given x.
Question (JP): 与えられた x に対し、数列 a_n = cos^n(x) の収束をシミュレートせよ。
Hint: Use numpy.cos with repeated powers.
[Chapter 1 - Problem 16]
Question (EN): Implement a function that returns the limit of a recursive sequence a_{n+1} = 0.5*(a_n + 2/a_n), starting with a_0=1.
Question (JP): 再帰的数列 a_{n+1} = 0.5*(a_n + 2/a_n), a_0=1 の極限を求める関数を実装せよ。
Hint: Newton’s method for sqrt(2).
[Chapter 1 - Problem 17]
Question (EN): Write a Python function to generate the decimal expansion of π up to 50 digits.
Question (JP): π の小数展開を50桁まで生成する関数を書け。
Hint: Use mpmath library.
[Chapter 1 - Problem 18]
Question (EN): Approximate the golden ratio φ using the sequence a_n = F_{n+1}/F_n (F_n = Fibonacci numbers).
Question (JP): フィボナッチ数列を用いて黄金比 φ を近似せよ。
Hint: Use recursion or memoization.
[Chapter 1 - Problem 19]
Question (EN): Simulate and plot the logistic map x_{n+1} = r x_n (1 - x_n) for r=3.5, starting from x_0=0.5.
Question (JP): r=3.5, x_0=0.5 のロジスティック写像をシミュレーションし、プロットせよ。
Hint: Use matplotlib.
[Chapter 1 - Problem 20]
Question (EN): Write Python code to test whether the decimal expansion of 1/13 repeats, and find its period length.
Question (JP): 1/13 の小数展開が循環するか確認し、その循環の長さを求めよ。
Hint: Use modular arithmetic.
[Chapter 1 - Problem 21]
Question (EN): Implement a function that checks if a sequence is bounded above and below.
Question (JP): 数列が上に有界かつ下に有界であるかを判定する関数を実装せよ。
Hint: Use max() and min().
[Chapter 1 - Problem 22]
Question (EN): Write Python code to compute the nth root of a number using Newton’s method.
Question (JP): ニュートン法を用いて数の n 乗根を計算するコードを書け。
Hint: Iterative approach.
[Chapter 1 - Problem 23]
Question (EN): Write a program that simulates the convergence of the sequence a_n = (n+1)/n.
Question (JP): 数列 a_n = (n+1)/n の収束をシミュレーションするプログラムを書け。
Hint: Use numpy for large n.
[Chapter 1 - Problem 24]
Question (EN): Implement a Python function to approximate the decimal expansion of √3 up to 30 digits.
Question (JP): √3 の小数展開を30桁まで求める関数を実装せよ。
Hint: Use decimal.getcontext().
[Chapter 1 - Problem 25]
Question (EN): Simulate the convergence of the Cesàro mean of the sequence (-1)^n.
Question (JP): 数列 (-1)^n のセザロ平均の収束をシミュレートせよ。
Hint: Compute averages of partial sums.
[Chapter 2 - Problem 1]
Question (EN): Write Python code to numerically approximate the limit lim_{x→0} (sin(x)/x).
Question (JP): 極限 lim_{x→0} (sin(x)/x) を数値的に近似するPythonコードを書け。
Hint: Use numpy with small values of x.
[Chapter 2 - Problem 2]
Question (EN): Implement a program that plots the function f(x) = 1/x and discuss its discontinuity at x=0.
Question (JP): 関数 f(x) = 1/x をプロットし、x=0での不連続性を考察せよ。
Hint: Use matplotlib with domain restriction.
[Chapter 2 - Problem 3]
Question (EN): Write Python code to approximate lim_{x→∞} (1 + 1/x)^x.
Question (JP): 極限 lim_{x→∞} (1 + 1/x)^x を近似するPythonコードを書け。
Hint: Try increasing x values.
[Chapter 2 - Problem 4]
Question (EN): Create a program that checks if a given function is continuous at a given point numerically.
Question (JP): 関数が指定された点で連続かどうかを数値的に確認するプログラムを書け。
Hint: Evaluate left and right limits.
[Chapter 2 - Problem 5]
Question (EN): Plot the functions sin(x), cos(x), and tan(x) on the same graph.
Question (JP): 関数 sin(x), cos(x), tan(x) を同じグラフにプロットせよ。
Hint: Use numpy.linspace and matplotlib.
[Chapter 2 - Problem 6]
Question (EN): Write Python code to check whether the function f(x) = |x| is differentiable at x=0.
Question (JP): f(x) = |x| が x=0 で微分可能かどうかを確認するPythonコードを書け。
Hint: Use difference quotient.
[Chapter 2 - Problem 7]
Question (EN): Implement a function to compute numerical left and right limits of f(x) = 1/(x-2) at x=2.
Question (JP): f(x) = 1/(x-2) の x=2 における左極限と右極限を数値的に求めよ。
Hint: Approach 2 from both sides.
[Chapter 2 - Problem 8]
Question (EN): Plot the exponential function f(x) = exp(x) for -5 ≤ x ≤ 5.
Question (JP): 指数関数 f(x) = exp(x) を区間 -5 ≤ x ≤ 5 でプロットせよ。
Hint: Use numpy.exp.
[Chapter 2 - Problem 9]
Question (EN): Plot the logarithmic function f(x) = log(x) for 0 < x ≤ 5.
Question (JP): 対数関数 f(x) = log(x) を区間 0 < x ≤ 5 でプロットせよ。
Hint: Use numpy.log.
[Chapter 2 - Problem 10]
Question (EN): Write a Python function to approximate the limit of f(x) = (x^2 - 1)/(x - 1) as x → 1.
Question (JP): f(x) = (x^2 - 1)/(x - 1) の x→1 の極限を近似する関数を書け。
Hint: Simplify algebraically and compare.
[Chapter 2 - Problem 11]
Question (EN): Plot the function f(x) = floor(x) and discuss its discontinuities.
Question (JP): 関数 f(x) = floor(x) をプロットし、不連続点を考察せよ。
Hint: Use numpy.floor.
[Chapter 2 - Problem 12]
Question (EN): Implement a program to compute the left-hand and right-hand derivatives of f(x) = |x| at x=0.
Question (JP): f(x) = |x| の x=0 における左微分と右微分を求めるプログラムを書け。
Hint: Use finite differences.
[Chapter 2 - Problem 13]
Question (EN): Plot the function f(x) = x^2 sin(1/x) for -0.1 ≤ x ≤ 0.1 and examine continuity at 0.
Question (JP): 関数 f(x) = x^2 sin(1/x) を -0.1 ≤ x ≤ 0.1 でプロットし、x=0での連続性を調べよ。
Hint: Avoid x=0 directly.
[Chapter 2 - Problem 14]
Question (EN): Write Python code to test the Intermediate Value Theorem for f(x) = x^3 - x on [0,1].
Question (JP): f(x) = x^3 - x が区間 [0,1] で中間値の定理を満たすことを確認せよ。
Hint: Evaluate f at endpoints.
[Chapter 2 - Problem 15]
Question (EN): Plot the function f(x) = sin(1/x) for 0 < x ≤ 1 and describe its behavior near 0.
Question (JP): f(x) = sin(1/x) を 0 < x ≤ 1 でプロットし、0付近の挙動を考察せよ。
Hint: Use numpy.linspace with small step.
[Chapter 2 - Problem 16]
Question (EN): Write Python code to approximate lim_{x→0} (1 - cos(x))/x^2.
Question (JP): 極限 lim_{x→0} (1 - cos(x))/x^2 を近似するPythonコードを書け。
Hint: Use small x values.
[Chapter 2 - Problem 17]
Question (EN): Plot the function f(x) = tanh(x) and analyze its asymptotic behavior.
Question (JP): 関数 f(x) = tanh(x) をプロットし、その漸近挙動を分析せよ。
Hint: Use numpy.tanh.
[Chapter 2 - Problem 18]
Question (EN): Implement code to approximate lim_{x→∞} (ln(x)/x).
Question (JP): 極限 lim_{x→∞} (ln(x)/x) を近似するコードを書け。
Hint: Try large x values.
[Chapter 2 - Problem 19]
Question (EN): Plot f(x) = x^x for 0 < x ≤ 5 and examine its behavior near x=0.
Question (JP): f(x) = x^x を 0 < x ≤ 5 でプロットし、x=0付近の挙動を考察せよ。
Hint: Use numpy.power.
[Chapter 2 - Problem 20]
Question (EN): Write Python code to test continuity of f(x) = sin(x)/x at x=0 using numerical approximation.
Question (JP): f(x) = sin(x)/x の x=0 における連続性を数値的に確認せよ。
Hint: Use limits from both sides.
[Chapter 3 - Problem 1]
Question (EN): Write a Python function to compute the derivative of f(x) = x^2 numerically.
Question (JP): f(x) = x^2 の導関数を数値的に計算する関数を書け。
Hint: Use finite difference.
[Chapter 3 - Problem 2]
Question (EN): Implement code to approximate the derivative of f(x) = sin(x) at multiple points.
Question (JP): f(x) = sin(x) の導関数を複数の点で近似せよ。
Hint: Compare with cos(x).
[Chapter 3 - Problem 3]
Question (EN): Plot f(x) = |x| and its numerical derivative around x=0.
Question (JP): f(x) = |x| をプロットし、その数値微分を x=0 付近で描け。
Hint: Use central difference.
[Chapter 3 - Problem 4]
Question (EN): Write Python code to apply the product rule to f(x) = x^2 * sin(x) and compare with numerical derivative.
Question (JP): f(x) = x^2 * sin(x) に積の微分法を適用し、数値微分と比較せよ。
Hint: Use symbolic differentiation (sympy).
[Chapter 3 - Problem 5]
Question (EN): Implement Python code to test the chain rule for f(x) = sin(x^2).
Question (JP): f(x) = sin(x^2) に連鎖律を適用するPythonコードを書け。
Hint: Differentiate outer and inner functions.
[Chapter 3 - Problem 6]
Question (EN): Write Python code to approximate the derivative of f(x) = exp(x) at x=0.
Question (JP): f(x) = exp(x) の導関数を x=0 で近似せよ。
Hint: Compare with the exact derivative f'(0)=1.
[Chapter 3 - Problem 7]
Question (EN): Plot the function f(x) = ln(x) and approximate its derivative at several points.
Question (JP): f(x) = ln(x) をプロットし、いくつかの点で微分を近似せよ。
Hint: Use finite differences.
[Chapter 3 - Problem 8]
Question (EN): Write a program that approximates higher-order derivatives of f(x) = sin(x).
Question (JP): f(x) = sin(x) の高次導関数を近似するプログラムを書け。
Hint: Use repeated finite differences.
[Chapter 3 - Problem 9]
Question (EN): Implement code to approximate the derivative of f(x) = sqrt(x) at x=0.01, 0.1, 1, 10.
Question (JP): f(x) = √x の導関数を x=0.01, 0.1, 1, 10 で近似せよ。
Hint: Compare with f'(x)=1/(2√x).
[Chapter 3 - Problem 10]
Question (EN): Plot f(x) = tan(x) and check its derivative numerically around x=π/4.
Question (JP): f(x) = tan(x) をプロットし、x=π/4 付近で導関数を数値的に確認せよ。
Hint: Compare with f'(x)=sec^2(x).
[Chapter 3 - Problem 11]
Question (EN): Write Python code to verify L'Hôpital's rule for lim_{x→0} (sin(x)/x).
Question (JP): 極限 lim_{x→0} (sin(x)/x) に対してロピタルの定理を確認するPythonコードを書け。
Hint: Differentiate numerator and denominator.
[Chapter 3 - Problem 12]
Question (EN): Implement code to approximate lim_{x→0} (1-cos(x))/x^2 using L'Hôpital’s rule.
Question (JP): 極限 lim_{x→0} (1-cos(x))/x^2 をロピタルの定理で確認せよ。
Hint: Use derivatives of numerator and denominator.
[Chapter 3 - Problem 13]
Question (EN): Plot f(x) = (x^3 - 1)/(x-1) near x=1 and check the limit with L'Hôpital’s rule.
Question (JP): f(x) = (x^3 - 1)/(x-1) を x=1 付近でプロットし、ロピタルの定理で極限を確認せよ。
Hint: Apply derivative quotient.
[Chapter 3 - Problem 14]
Question (EN): Write Python code to implement numerical differentiation for noisy data.
Question (JP): ノイズを含むデータに対して数値微分を実装するPythonコードを書け。
Hint: Use smoothing before differentiation.
[Chapter 3 - Problem 15]
Question (EN): Implement symbolic differentiation of f(x) = (x^2+1)/(x-1) using sympy.
Question (JP): f(x) = (x^2+1)/(x-1) を sympy を使って記号微分せよ。
Hint: Use sympy.diff.
[Chapter 3 - Problem 16]
Question (EN): Write code to approximate the Taylor expansion of f(x)=exp(x) up to order 5.
Question (JP): f(x) = exp(x) のテイラー展開を5次まで近似するコードを書け。
Hint: Use series expansion.
[Chapter 3 - Problem 17]
Question (EN): Plot f(x)=sin(x) and its Taylor polynomial approximation around x=0.
Question (JP): f(x) = sin(x) とその x=0 周りのテイラー多項式近似をプロットせよ。
Hint: Use numpy.polynomial.
[Chapter 3 - Problem 18]
Question (EN): Implement code to compare the Taylor approximation of cos(x) up to 2nd, 4th, and 6th order.
Question (JP): cos(x) のテイラー近似（2次, 4次, 6次）を比較せよ。
Hint: Plot all together.
[Chapter 3 - Problem 19]
Question (EN): Write Python code to test the error of Taylor approximation for exp(x) at x=1.
Question (JP): exp(x) のテイラー展開近似の誤差を x=1 で調べよ。
Hint: Compute actual value minus approximation.
[Chapter 3 - Problem 20]
Question (EN): Implement code to approximate derivatives using automatic differentiation libraries.
Question (JP): 自動微分ライブラリを使って導関数を近似せよ。
Hint: Try jax or autograd.
[Chapter 3 - Problem 21]
Question (EN): Write a function that verifies the Mean Value Theorem numerically for f(x) = x^2 on [0,2].
Question (JP): f(x) = x^2 が区間 [0,2] で平均値の定理を満たすことを数値的に確認せよ。
Hint: Check derivative at some c ∈ (0,2).
[Chapter 3 - Problem 22]
Question (EN): Plot the tangent line to f(x)=ln(x) at x=1.
Question (JP): f(x)=ln(x) の接線を x=1 で描け。
Hint: Use slope = f’(1).
[Chapter 3 - Problem 23]
Question (EN): Write Python code to approximate the derivative of f(x)=sin(x^2) using the chain rule.
Question (JP): f(x)=sin(x^2) の導関数を連鎖律で求め、数値微分と比較せよ。
Hint: f’(x)=2x cos(x^2).
[Chapter 3 - Problem 24]
Question (EN): Plot f(x)=1/x near x=0 and discuss differentiability.
Question (JP): f(x)=1/x を x=0 付近でプロットし、微分可能性を考察せよ。
Hint: Function undefined at x=0.
[Chapter 3 - Problem 25]
Question (EN): Write code to approximate second derivatives of f(x)=x^3 numerically.
Question (JP): f(x)=x^3 の二階導関数を数値的に近似するコードを書け。
Hint: Use central difference twice.
[Chapter 4 - Problem 1]
Question (EN): Write Python code to approximate the area under f(x)=x from 0 to 1 using rectangles.
Question (JP): f(x)=x の 0〜1 の面積を長方形近似で求めるPythonコードを書け。
Hint: Use Riemann sums.
[Chapter 4 - Problem 2]
Question (EN): Implement numerical integration for f(x)=x^2 from 0 to 2 using the trapezoidal rule.
Question (JP): f(x)=x^2 の 0〜2 の積分を台形公式で近似せよ。
Hint: Use numpy.trapz.
[Chapter 4 - Problem 3]
Question (EN): Write Python code to compute ∫_0^π sin(x) dx numerically.
Question (JP): 積分 ∫_0^π sin(x) dx を数値的に求めよ。
Hint: Compare with exact value 2.
[Chapter 4 - Problem 4]
Question (EN): Plot convergence of trapezoidal approximations for ∫_0^1 exp(x) dx.
Question (JP): ∫_0^1 exp(x) dx の台形公式近似の収束をプロットせよ。
Hint: Increase number of subintervals.
[Chapter 4 - Problem 5]
Question (EN): Write Python code to approximate ∫_0^1 1/(1+x^2) dx and compare with arctan(1).
Question (JP): ∫_0^1 1/(1+x^2) dx を近似し、arctan(1) と比較せよ。
Hint: Exact = π/4.
[Chapter 4 - Problem 6]
Question (EN): Implement Simpson’s rule for ∫_0^2 x^2 dx.
Question (JP): ∫_0^2 x^2 dx をシンプソン則で計算せよ。
Hint: Compare with exact 8/3.
[Chapter 4 - Problem 7]
Question (EN): Write Python code to integrate f(x)=cos(x) on [0,π/2] using scipy.integrate.quad.
Question (JP): f(x)=cos(x) を [0,π/2] で積分せよ（scipy.integrate.quad を用いる）。
Hint: Compare with exact value 1.
[Chapter 4 - Problem 8]
Question (EN): Plot error convergence of trapezoidal vs Simpson’s rule for ∫_0^1 exp(x).
Question (JP): ∫_0^1 exp(x) の積分誤差収束を台形公式とシンプソン則で比較せよ。
Hint: Use log-log plot.
[Chapter 4 - Problem 9]
Question (EN): Write Python code to compute ∫_0^∞ exp(-x) dx numerically.
Question (JP): ∫_0^∞ exp(-x) dx を数値的に近似せよ。
Hint: Exact = 1.
[Chapter 4 - Problem 10]
Question (EN): Approximate ∫_1^∞ 1/x^2 dx numerically and compare with exact 1.
Question (JP): ∫_1^∞ 1/x^2 dx を近似し、正確値1と比較せよ。
Hint: Try large cutoff.
[Chapter 4 - Problem 11]
Question (EN): Plot partial integrals of ∫_0^∞ sin(x)/x dx and observe convergence.
Question (JP): ∫_0^∞ sin(x)/x dx の部分積分をプロットし、収束を確認せよ。
Hint: This is Dirichlet integral.
[Chapter 4 - Problem 12]
Question (EN): Write Python code to compute ∫_0^1 sqrt(1-x^2) dx (area of quarter circle).
Question (JP): ∫_0^1 √(1-x^2) dx を計算せよ（1/4円の面積）。
Hint: Compare with π/4.
[Chapter 4 - Problem 13]
Question (EN): Approximate ∫_0^π |sin(x)| dx numerically.
Question (JP): ∫_0^π |sin(x)| dx を数値的に求めよ。
Hint: Result = 2.
[Chapter 4 - Problem 14]
Question (EN): Implement Monte Carlo integration for ∫_0^1 x^2 dx.
Question (JP): ∫_0^1 x^2 dx をモンテカルロ積分で近似せよ。
Hint: Use random numbers.
[Chapter 4 - Problem 15]
Question (EN): Write Python code to approximate ∫_0^1 4/(1+x^2) dx using Monte Carlo method.
Question (JP): ∫_0^1 4/(1+x^2) dx をモンテカルロ法で近似せよ。
Hint: Estimate π.
[Chapter 4 - Problem 16]
Question (EN): Implement Gaussian quadrature for ∫_0^1 x^3 dx.
Question (JP): ∫_0^1 x^3 dx をガウス求積で計算せよ。
Hint: Compare with exact 1/4.
[Chapter 4 - Problem 17]
Question (EN): Write code to approximate ∫_0^∞ exp(-x^2) dx numerically.
Question (JP): ∫_0^∞ exp(-x^2) dx を近似せよ。
Hint: Compare with √π/2.
[Chapter 4 - Problem 18]
Question (EN): Plot convergence of Monte Carlo integration for ∫_0^1 sin(x) dx.
Question (JP): ∫_0^1 sin(x) dx のモンテカルロ積分の収束をプロットせよ。
Hint: Increase sample size.
[Chapter 4 - Problem 19]
Question (EN): Write Python code to compute ∫_0^∞ x exp(-x) dx.
Question (JP): ∫_0^∞ x exp(-x) dx を計算せよ。
Hint: Exact = 1.
[Chapter 4 - Problem 20]
Question (EN): Approximate ∫_0^∞ e^{-ax} dx for a=2 numerically.
Question (JP): a=2 のとき ∫_0^∞ e^{-ax} dx を近似せよ。
Hint: Exact = 1/a.
[Chapter 4 - Problem 21]
Question (EN): Implement Simpson’s rule for ∫_0^π cos(x) dx.
Question (JP): ∫_0^π cos(x) dx をシンプソン則で計算せよ。
Hint: Exact = 0.
[Chapter 4 - Problem 22]
Question (EN): Write Python code to compute ∫_{-1}^1 1/(1+x^2) dx.
Question (JP): ∫_{-1}^1 1/(1+x^2) dx を計算せよ。
Hint: Compare with 2 arctan(1).
[Chapter 4 - Problem 23]
Question (EN): Approximate ∫_0^1 log(x) dx numerically.
Question (JP): ∫_0^1 log(x) dx を数値的に近似せよ。
Hint: Exact = -1.
[Chapter 4 - Problem 24]
Question (EN): Plot integrand f(x)=sin(x)/x and compute ∫_0^{10} f(x) dx.
Question (JP): f(x)=sin(x)/x をプロットし、∫_0^{10} f(x) dx を計算せよ。
Hint: Use scipy.integrate.quad.
[Chapter 4 - Problem 25]
Question (EN): Write Python code to compute improper integral ∫_0^∞ e^{-x}/(1+x) dx.
Question (JP): ∫_0^∞ e^{-x}/(1+x) dx を数値的に計算せよ。
Hint: Use scipy.quad with limit.
[Chapter 4 - Problem 26]
Question (EN): Implement Monte Carlo estimation of area of a unit circle using random points.
Question (JP): 乱数を使って単位円の面積をモンテカルロ法で推定せよ。
Hint: Estimate π.
[Chapter 4 - Problem 27]
Question (EN): Compute ∫_0^∞ x^2 e^{-x} dx numerically.
Question (JP): ∫_0^∞ x^2 e^{-x} dx を数値的に求めよ。
Hint: Exact = 2.
[Chapter 4 - Problem 28]
Question (EN): Approximate ∫_0^1 sqrt(x) dx using trapezoidal rule.
Question (JP): ∫_0^1 √x dx を台形公式で近似せよ。
Hint: Exact = 2/3.
[Chapter 4 - Problem 29]
Question (EN): Plot convergence of Simpson’s rule for ∫_0^1 1/(1+x) dx.
Question (JP): ∫_0^1 1/(1+x) dx のシンプソン則近似の収束をプロットせよ。
Hint: Exact = ln(2).
[Chapter 4 - Problem 30]
Question (EN): Write Python code to compare trapezoidal, Simpson, and Monte Carlo integration for ∫_0^1 exp(x) dx.
Question (JP): ∫_0^1 exp(x) dx を台形公式・シンプソン則・モンテカルロ法で比較せよ。
Hint: Compare errors.
[Chapter 5 - Problem 1]
Question (EN): Write Python code to generate random points in 2D Euclidean space and compute their distances from the origin.
Question (JP): 2次元ユークリッド空間で乱数点を生成し、原点からの距離を計算せよ。
Hint: Use sqrt(x^2+y^2).
[Chapter 5 - Problem 2]
Question (EN): Plot the 3D surface of f(x,y) = x^2 + y^2.
Question (JP): f(x,y) = x^2 + y^2 の3D曲面をプロットせよ。
Hint: Use matplotlib Axes3D.
[Chapter 5 - Problem 3]
Question (EN): Write code to compute f(x,y) = sqrt(x^2 + y^2) for a grid of points.
Question (JP): 格子点に対して f(x,y) = √(x^2 + y^2) を計算せよ。
Hint: Use numpy.meshgrid.
[Chapter 5 - Problem 4]
Question (EN): Plot contour lines of f(x,y)=x^2 - y^2.
Question (JP): f(x,y) = x^2 - y^2 の等高線をプロットせよ。
Hint: Use matplotlib.contour.
[Chapter 5 - Problem 5]
Question (EN): Write Python code to compute f(x,y)=sin(x)cos(y) over -π ≤ x,y ≤ π.
Question (JP): f(x,y)=sin(x)cos(y) を -π ≤ x,y ≤ π で計算せよ。
Hint: Use numpy broadcasting.
[Chapter 5 - Problem 6]
Question (EN): Implement a function that checks if a point belongs to the unit circle x^2+y^2≤1.
Question (JP): 点が単位円 x^2+y^2≤1 に属するかを判定する関数を書け。
Hint: Return True/False.
[Chapter 5 - Problem 7]
Question (EN): Plot 3D surface of f(x,y)=exp(-(x^2+y^2)).
Question (JP): f(x,y)=exp(-(x^2+y^2)) の3D曲面を描け。
Hint: Use matplotlib surface plot.
[Chapter 5 - Problem 8]
Question (EN): Write Python code to compute distance between two random points in 3D Euclidean space.
Question (JP): 3次元ユークリッド空間における2点間距離を計算するコードを書け。
Hint: Use sqrt((x1-x2)^2+(y1-y2)^2+(z1-z2)^2).
[Chapter 5 - Problem 9]
Question (EN): Plot contour map of f(x,y)=sin(sqrt(x^2+y^2)).
Question (JP): f(x,y)=sin(√(x^2+y^2)) の等高線を描け。
Hint: Use numpy.sqrt with meshgrid.
[Chapter 5 - Problem 10]
Question (EN): Write Python code to compute maximum and minimum values of f(x,y)=x^2+y^2 for x,y∈[-1,1].
Question (JP): f(x,y)=x^2+y^2 の最大値・最小値を [-1,1]×[-1,1] で求めよ。
Hint: Use numpy.max and numpy.min.
[Chapter 5 - Problem 11]
Question (EN): Plot f(x,y)=xy and discuss its symmetry properties.
Question (JP): f(x,y)=xy をプロットし、その対称性を考察せよ。
Hint: Odd/even symmetry.
[Chapter 5 - Problem 12]
Question (EN): Write Python code to compute the gradient numerically for f(x,y)=x^2+y^2.
Question (JP): f(x,y)=x^2+y^2 の勾配を数値的に計算せよ。
Hint: Use finite differences.
[Chapter 5 - Problem 13]
Question (EN): Plot level curves of f(x,y)=x^2+y^2 and compare with circles.
Question (JP): f(x,y)=x^2+y^2 の等高線を描き、円と比較せよ。
Hint: Concentric circles.
[Chapter 5 - Problem 14]
Question (EN): Write Python code to evaluate f(x,y)=x^2y - y^3 at a set of random points.
Question (JP): f(x,y)=x^2y - y^3 をランダムな点で計算せよ。
Hint: Use numpy.random.
[Chapter 5 - Problem 15]
Question (EN): Plot the function f(x,y)=1/(1+x^2+y^2) as a 3D surface.
Question (JP): f(x,y)=1/(1+x^2+y^2) を3D曲面でプロットせよ。
Hint: Use matplotlib surface plot.
[Chapter 6 - Problem 1]
Question (EN): Write Python code to compute partial derivatives of f(x,y)=x^2+y^2 at (1,2).
Question (JP): f(x,y)=x^2+y^2 の (1,2) における偏微分を計算せよ。
Hint: Use sympy.diff.
[Chapter 6 - Problem 2]
Question (EN): Implement numerical approximation of ∂f/∂x for f(x,y)=sin(xy) at (1,1).
Question (JP): f(x,y)=sin(xy) の (1,1) における ∂f/∂x を数値的に近似せよ。
Hint: Use finite difference.
[Chapter 6 - Problem 3]
Question (EN): Plot gradient field of f(x,y)=x^2+y^2.
Question (JP): f(x,y)=x^2+y^2 の勾配ベクトル場をプロットせよ。
Hint: Use quiver plot.
[Chapter 6 - Problem 4]
Question (EN): Write Python code to compute Jacobian of f(x,y)=(x^2,xy).
Question (JP): f(x,y)=(x^2,xy) のヤコビ行列を計算せよ。
Hint: Use sympy.Matrix.jacobian.
[Chapter 6 - Problem 5]
Question (EN): Implement numerical gradient descent for f(x,y)=x^2+y^2.
Question (JP): f(x,y)=x^2+y^2 に対して数値的な勾配降下法を実装せよ。
Hint: Update rule (x,y) ← (x,y) - η∇f.
[Chapter 6 - Problem 6]
Question (EN): Write code to compute Hessian matrix of f(x,y)=x^2y+y^2.
Question (JP): f(x,y)=x^2y+y^2 のヘッセ行列を計算せよ。
Hint: Use sympy.hessian.
[Chapter 6 - Problem 7]
Question (EN): Plot directional derivatives of f(x,y)=x^2+y^2 at (1,1).
Question (JP): f(x,y)=x^2+y^2 の (1,1) における方向微分をプロットせよ。
Hint: Use ∇f · u.
[Chapter 6 - Problem 8]
Question (EN): Implement Python code to check differentiability of f(x,y)=|xy| at (0,0).
Question (JP): f(x,y)=|xy| が (0,0) で微分可能かを確認せよ。
Hint: Compare partial derivatives.
[Chapter 6 - Problem 9]
Question (EN): Write Python code to approximate ∂f/∂x for f(x,y)=ln(x^2+y^2) at (1,1).
Question (JP): f(x,y)=ln(x^2+y^2) の (1,1) における ∂f/∂x を近似せよ。
Hint: Use finite difference.
[Chapter 6 - Problem 10]
Question (EN): Plot the gradient vector field of f(x,y)=sin(x)+cos(y).
Question (JP): f(x,y)=sin(x)+cos(y) の勾配ベクトル場を描け。
Hint: Use quiver.
[Chapter 6 - Problem 11]
Question (EN): Write code to compute critical points of f(x,y)=x^3-3xy^2.
Question (JP): f(x,y)=x^3-3xy^2 の停留点を求めよ。
Hint: Solve ∂f/∂x=0, ∂f/∂y=0.
[Chapter 6 - Problem 12]
Question (EN): Implement code to classify critical points of f(x,y)=x^2-y^2.
Question (JP): f(x,y)=x^2-y^2 の停留点を分類せよ。
Hint: Use Hessian determinant test.
[Chapter 6 - Problem 13]
Question (EN): Plot the tangent plane of f(x,y)=x^2+y^2 at (1,1).
Question (JP): f(x,y)=x^2+y^2 の (1,1) における接平面を描け。
Hint: z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).
[Chapter 6 - Problem 14]
Question (EN): Write Python code to approximate gradient numerically for f(x,y)=exp(xy).
Question (JP): f(x,y)=exp(xy) の勾配を数値的に近似せよ。
Hint: Use central differences.
[Chapter 6 - Problem 15]
Question (EN): Implement code to compute ∂^2f/∂x∂y for f(x,y)=sin(xy).
Question (JP): f(x,y)=sin(xy) の混合二階偏微分 ∂^2f/∂x∂y を求めよ。
Hint: Use sympy.diff.
[Chapter 6 - Problem 16]
Question (EN): Write Python code to compute divergence of F(x,y)=(x,y).
Question (JP): F(x,y)=(x,y) の発散を計算せよ。
Hint: div F = ∂x/∂x+∂y/∂y.
[Chapter 6 - Problem 17]
Question (EN): Implement code to compute curl of F(x,y)=(y,-x).
Question (JP): F(x,y)=(y,-x) の回転を計算せよ。
Hint: curl = ∂Q/∂x - ∂P/∂y.
[Chapter 6 - Problem 18]
Question (EN): Write code to compute Laplacian of f(x,y)=x^2+y^2.
Question (JP): f(x,y)=x^2+y^2 のラプラシアンを計算せよ。
Hint: Δf=f_xx+f_yy.
[Chapter 6 - Problem 19]
Question (EN): Plot gradient descent path for f(x,y)=x^2+y^2 starting at (2,2).
Question (JP): f(x,y)=x^2+y^2 の勾配降下法の経路を (2,2) から描け。
Hint: Iterate updates.
[Chapter 6 - Problem 20]
Question (EN): Write code to compute total derivative of f(x,y)=x^2+y with respect to t where x=t, y=t^2.
Question (JP): x=t, y=t^2 のとき f(x,y)=x^2+y の全微分を計算せよ。
Hint: Use chain rule.
[Chapter 6 - Problem 21]
Question (EN): Implement code to check implicit function y(x) from x^2+y^2=1 at (0,1).
Question (JP): x^2+y^2=1 から (0,1) での陰関数 y(x) を確認せよ。
Hint: Use ∂F/∂y≠0.
[Chapter 6 - Problem 22]
Question (EN): Write code to compute Jacobian determinant of transformation (u=x+y, v=x-y).
Question (JP): 変換 (u=x+y, v=x-y) のヤコビ行列式を求めよ。
Hint: det = -2.
[Chapter 6 - Problem 23]
Question (EN): Implement code to compute gradient of f(x,y,z)=x^2+y^2+z^2.
Question (JP): f(x,y,z)=x^2+y^2+z^2 の勾配を計算せよ。
Hint: ∇f=(2x,2y,2z).
[Chapter 6 - Problem 24]
Question (EN): Write code to compute directional derivative of f(x,y)=xy at (1,1) in direction (1,1).
Question (JP): f(x,y)=xy の (1,1) における方向 (1,1) の方向微分を求めよ。
Hint: ∇f·u.
[Chapter 6 - Problem 25]
Question (EN): Plot vector field F(x,y)=(x^2,y^2) and compute divergence numerically.
Question (JP): F(x,y)=(x^2,y^2) をプロットし、発散を数値的に求めよ。
Hint: Use finite difference for ∂/∂x and ∂/∂y.
[Chapter 7 - Problem 1]
Question (EN): Write Python code to compute the double integral ∫∫_D 1 dA where D=[0,1]×[0,1].
Question (JP): D=[0,1]×[0,1] 上の二重積分 ∫∫_D 1 dA を計算せよ。
Hint: Result = 1.
[Chapter 7 - Problem 2]
Question (EN): Implement code to approximate ∫∫_D (x+y) dA where D=[0,1]×[0,1].
Question (JP): D=[0,1]×[0,1] 上で ∫∫_D (x+y) dA を近似せよ。
Hint: Exact = 1.
[Chapter 7 - Problem 3]
Question (EN): Plot the region D={(x,y): x^2+y^2≤1} and compute ∫∫_D 1 dA.
Question (JP): 領域 D={(x,y): x^2+y^2≤1} を描き、∫∫_D 1 dA を計算せよ。
Hint: Area = π.
[Chapter 7 - Problem 4]
Question (EN): Write Python code to compute ∫∫_D x^2+y^2 dA for D=unit disk.
Question (JP): 単位円領域 D 上で ∫∫_D (x^2+y^2) dA を計算せよ。
Hint: Use polar coordinates.
[Chapter 7 - Problem 5]
Question (EN): Implement Monte Carlo integration for ∫∫_D sin(x)cos(y) dA on D=[0,π]×[0,π].
Question (JP): D=[0,π]×[0,π] 上で ∫∫_D sin(x)cos(y) dA をモンテカルロ法で近似せよ。
Hint: Use random.uniform.
[Chapter 7 - Problem 6]
Question (EN): Plot 3D surface of integrand f(x,y)=exp(-(x^2+y^2)) and approximate ∫∫ f(x,y) dA over [-2,2]^2.
Question (JP): f(x,y)=exp(-(x^2+y^2)) の3D曲面を描き、[-2,2]^2 上の二重積分を近似せよ。
Hint: Compare with π.
[Chapter 7 - Problem 7]
Question (EN): Write Python code to compute ∫∫_D xy dA where D=[0,1]×[0,1].
Question (JP): D=[0,1]×[0,1] 上で ∫∫_D xy dA を計算せよ。
Hint: Exact = 1/4.
[Chapter 7 - Problem 8]
Question (EN): Implement code to compute ∫∫_D (x^2+y^2) dA where D is triangle (0,0),(1,0),(0,1).
Question (JP): 三角形 D=(0,0),(1,0),(0,1) 上で ∫∫_D (x^2+y^2) dA を計算せよ。
Hint: Use double integral bounds.
[Chapter 7 - Problem 9]
Question (EN): Write code to compute ∫∫_D 1 dA where D={(x,y):0≤y≤x,0≤x≤1}.
Question (JP): D={(x,y):0≤y≤x,0≤x≤1} 上で ∫∫_D 1 dA を計算せよ。
Hint: Area = 1/2.
[Chapter 7 - Problem 10]
Question (EN): Compute ∫∫_D x dA for D=[0,1]×[0,1].
Question (JP): D=[0,1]×[0,1] 上で ∫∫_D x dA を計算せよ。
Hint: Exact = 1/2.
[Chapter 7 - Problem 11]
Question (EN): Implement polar coordinates integration for ∫∫_D r^2 dA where D=unit disk.
Question (JP): 単位円領域 D に対し、極座標で ∫∫_D r^2 dA を計算せよ。
Hint: Integral = π/2.
[Chapter 7 - Problem 12]
Question (EN): Plot integrand f(x,y)=1/(1+x^2+y^2) and approximate ∫∫ over [-1,1]^2.
Question (JP): f(x,y)=1/(1+x^2+y^2) を描き、[-1,1]^2 上で二重積分を近似せよ。
Hint: Use scipy.integrate.dblquad.
[Chapter 7 - Problem 13]
Question (EN): Write code to compute ∫∫_D (x+y) dA for triangular region (0,0),(1,0),(0,1).
Question (JP): 三角形 D=(0,0),(1,0),(0,1) 上で ∫∫_D (x+y) dA を計算せよ。
Hint: Symmetric result.
[Chapter 7 - Problem 14]
Question (EN): Implement Monte Carlo estimation for area of ellipse x^2/4+y^2≤1.
Question (JP): 楕円 x^2/4+y^2≤1 の面積をモンテカルロ法で推定せよ。
Hint: Exact area = 2π.
[Chapter 7 - Problem 15]
Question (EN): Write Python code to compute ∫∫_D e^{-(x^2+y^2)} dA over entire plane.
Question (JP): 平面全体で ∫∫ e^{-(x^2+y^2)} dA を計算せよ。
Hint: Exact = π.
[Chapter 7 - Problem 16]
Question (EN): Compute ∫∫_D (x^2-y^2) dA where D=[-1,1]×[-1,1].
Question (JP): D=[-1,1]×[-1,1] 上で ∫∫_D (x^2-y^2) dA を計算せよ。
Hint: Symmetry simplifies.
[Chapter 7 - Problem 17]
Question (EN): Write code to approximate ∫∫_D sin(x^2+y^2) dA over [-1,1]^2.
Question (JP): ∫∫_D sin(x^2+y^2) dA を [-1,1]^2 上で近似せよ。
Hint: Use grid approximation.
[Chapter 7 - Problem 18]
Question (EN): Implement triple integral ∭_D 1 dV where D=[0,1]×[0,1]×[0,1].
Question (JP): 立方体 D=[0,1]^3 上で ∭_D 1 dV を計算せよ。
Hint: Result = 1.
[Chapter 7 - Problem 19]
Question (EN): Write Python code to compute volume of sphere of radius 1 using triple integral.
Question (JP): 半径1の球の体積を三重積分で求めよ。
Hint: Exact = 4π/3.
[Chapter 7 - Problem 20]
Question (EN): Compute ∭_D (x^2+y^2+z^2) dV for unit ball.
Question (JP): 単位球領域 D で ∭_D (x^2+y^2+z^2) dV を計算せよ。
Hint: Use spherical coordinates.
[Chapter 8 - Problem 1]
Question (EN): Write Python code to compute the partial sums of the harmonic series ∑ 1/n up to n=1000.
Question (JP): 調和級数 ∑ 1/n の部分和を n=1000 まで計算せよ。
Hint: Use numpy.cumsum.
[Chapter 8 - Problem 2]
Question (EN): Plot convergence of the geometric series ∑ (1/2)^n up to n=20.
Question (JP): 等比級数 ∑ (1/2)^n の部分和の収束を n=20 までプロットせよ。
Hint: Compare with exact sum 1.
[Chapter 8 - Problem 3]
Question (EN): Write Python code to test convergence of the alternating harmonic series ∑ (-1)^(n+1)/n.
Question (JP): 交代調和級数 ∑ (-1)^(n+1)/n の収束を確認せよ。
Hint: Use partial sums.
[Chapter 8 - Problem 4]
Question (EN): Implement code to approximate ∑ 1/n^2 up to n=1000 and compare with π^2/6.
Question (JP): ∑ 1/n^2 を n=1000 まで近似し、π^2/6 と比較せよ。
Hint: Use numpy.
[Chapter 8 - Problem 5]
Question (EN): Plot partial sums of ∑ (-1)^n/n^2 and observe convergence.
Question (JP): ∑ (-1)^n/n^2 の部分和をプロットし、収束を観察せよ。
Hint: Use matplotlib.
[Chapter 8 - Problem 6]
Question (EN): Write Python code to compute Maclaurin series of exp(x) up to 6th order and plot.
Question (JP): exp(x) のマクローリン展開を6次まで計算し、プロットせよ。
Hint: Use sympy.series.
[Chapter 8 - Problem 7]
Question (EN): Implement code to approximate sin(x) using its Taylor series up to order 7.
Question (JP): sin(x) を7次のテイラー展開で近似せよ。
Hint: Compare with numpy.sin.
[Chapter 8 - Problem 8]
Question (EN): Plot approximation of cos(x) using Taylor polynomials of degree 2,4,6.
Question (JP): cos(x) の2次・4次・6次テイラー近似を比較してプロットせよ。
Hint: Overlay graphs.
[Chapter 8 - Problem 9]
Question (EN): Write Python code to approximate log(1+x) using its series expansion.
Question (JP): log(1+x) を級数展開で近似するコードを書け。
Hint: Use alternating series.
[Chapter 8 - Problem 10]
Question (EN): Plot approximation of arctan(x) using its Maclaurin series up to 7th order.
Question (JP): arctan(x) のマクローリン展開（7次）近似をプロットせよ。
Hint: Compare with numpy.arctan.
[Chapter 8 - Problem 11]
Question (EN): Write Python code to test radius of convergence of ∑ x^n/n.
Question (JP): 級数 ∑ x^n/n の収束半径を確認せよ。
Hint: Ratio test.
[Chapter 8 - Problem 12]
Question (EN): Implement code to compute ∑_{n=0}^{10} (x^n)/(n!) for x=1 and compare with exp(1).
Question (JP): x=1 のとき ∑_{n=0}^{10} (x^n)/(n!) を計算し、exp(1) と比較せよ。
Hint: Compare errors.
[Chapter 8 - Problem 13]
Question (EN): Plot approximation of sinh(x) using its series expansion.
Question (JP): sinh(x) の級数展開近似をプロットせよ。
Hint: Use odd powers.
[Chapter 8 - Problem 14]
Question (EN): Write Python code to compute partial sums of ∑ (-1)^n/(2n+1) and compare with π/4.
Question (JP): ∑ (-1)^n/(2n+1) の部分和を計算し、π/4 と比較せよ。
Hint: Leibniz series for π.
[Chapter 8 - Problem 15]
Question (EN): Implement code to test convergence of ∑ 1/n^p for different values of p.
Question (JP): ∑ 1/n^p が p の値により収束/発散することを確認せよ。
Hint: p>1 converges.
[Chapter 8 - Problem 16]
Question (EN): Write Python code to approximate ζ(2)=π^2/6 using partial sums.
Question (JP): ζ(2)=π^2/6 を部分和で近似せよ。
Hint: Use numpy.arange.
[Chapter 8 - Problem 17]
Question (EN): Plot approximation of e^x using partial sums up to n=5,10,20.
Question (JP): e^x の部分和近似を n=5,10,20 でプロットせよ。
Hint: Compare with numpy.exp.
[Chapter 8 - Problem 18]
Question (EN): Write code to approximate ∑ x^n for |x|<1 and compare with formula 1/(1-x).
Question (JP): |x|<1 のとき ∑ x^n を近似し、1/(1-x) と比較せよ。
Hint: Geometric series.
[Chapter 8 - Problem 19]
Question (EN): Implement Python code to approximate Fourier series of f(x)=x on [-π,π].
Question (JP): f(x)=x のフーリエ級数を [-π,π] で近似せよ。
Hint: Use sine terms.
[Chapter 8 - Problem 20]
Question (EN): Plot approximation of square wave using Fourier series with 5,10,50 terms.
Question (JP): 方形波をフーリエ級数で近似し、5,10,50項で比較せよ。
Hint: Observe Gibbs phenomenon.
[Chapter 9 - Problem 1]
Question (EN): Solve dy/dx = y with initial condition y(0)=1 using Python.
Question (JP): 微分方程式 dy/dx = y, y(0)=1 をPythonで解け。
Hint: Exact solution y=e^x.
[Chapter 9 - Problem 2]
Question (EN): Implement Euler’s method for dy/dx = -y, y(0)=1 with step size h=0.1.
Question (JP): dy/dx = -y, y(0)=1 をオイラー法で解け（刻み幅 h=0.1）。
Hint: Compare with e^{-x}.
[Chapter 9 - Problem 3]
Question (EN): Write Python code to solve dy/dx = x numerically with y(0)=0.
Question (JP): dy/dx = x, y(0)=0 を数値的に解け。
Hint: Exact solution y=x^2/2.
[Chapter 9 - Problem 4]
Question (EN): Plot solution of dy/dx = cos(x), y(0)=0 on [0,π].
Question (JP): dy/dx=cos(x), y(0)=0 の解を [0,π] でプロットせよ。
Hint: Exact y=sin(x).
[Chapter 9 - Problem 5]
Question (EN): Solve numerically dy/dx = -2y with y(0)=1 using scipy.integrate.solve_ivp.
Question (JP): dy/dx=-2y, y(0)=1 を solve_ivp で数値解を求めよ。
Hint: Compare with e^{-2x}.
[Chapter 9 - Problem 6]
Question (EN): Implement code to solve dy/dx = x+y with y(0)=1 numerically.
Question (JP): dy/dx = x+y, y(0)=1 を数値的に解け。
Hint: Linear ODE.
[Chapter 9 - Problem 7]
Question (EN): Write Python code to approximate solution of dy/dx = y^2, y(0)=1.
Question (JP): dy/dx = y^2, y(0)=1 の解を近似せよ。
Hint: Exact y=1/(1-x).
[Chapter 9 - Problem 8]
Question (EN): Plot slope field of dy/dx = x-y on [-2,2]×[-2,2].
Question (JP): dy/dx = x-y の方向場を [-2,2]×[-2,2] に描け。
Hint: Use quiver.
[Chapter 9 - Problem 9]
Question (EN): Solve second-order ODE y''+y=0 with y(0)=0, y'(0)=1 numerically.
Question (JP): y''+y=0, y(0)=0, y'(0)=1 を数値的に解け。
Hint: Exact y=sin(x).
[Chapter 9 - Problem 10]
Question (EN): Implement Python code to solve y''-y=0 with y(0)=1, y'(0)=0.
Question (JP): y''-y=0, y(0)=1, y'(0)=0 をPythonで解け。
Hint: Exact y=cosh(x).
[Chapter 9 - Problem 11]
Question (EN): Write Python code to solve logistic equation dy/dx = y(1-y), y(0)=0.1.
Question (JP): ロジスティック方程式 dy/dx=y(1-y), y(0)=0.1 を解け。
Hint: Exact y=1/(1+9e^{-x}).
[Chapter 9 - Problem 12]
Question (EN): Implement numerical solution of dy/dx = sin(x+y).
Question (JP): dy/dx = sin(x+y) を数値的に解け。
Hint: Use solve_ivp.
[Chapter 9 - Problem 13]
Question (EN): Plot phase portrait of system dx/dt=y, dy/dt=-x.
Question (JP): dx/dt=y, dy/dt=-x の相平面を描け。
Hint: Circular trajectories.
[Chapter 9 - Problem 14]
Question (EN): Write code to solve system dx/dt=x, dy/dt=-y with initial (1,1).
Question (JP): dx/dt=x, dy/dt=-y, 初期条件 (1,1) を解け。
Hint: Exact x=e^t, y=e^{-t}.
[Chapter 9 - Problem 15]
Question (EN): Implement Runge-Kutta 4th order method for dy/dx = -y, y(0)=1.
Question (JP): dy/dx=-y, y(0)=1 を4次ルンゲ＝クッタ法で解け。
Hint: Compare with e^{-x}.
[Chapter 9 - Problem 16]
Question (EN): Write Python code to compute numerical solution of y''+2y'+y=0 with y(0)=1,y'(0)=0.
Question (JP): y''+2y'+y=0, y(0)=1, y'(0)=0 を数値的に解け。
Hint: Exact y=e^{-x}.
[Chapter 9 - Problem 17]
Question (EN): Plot solution curves of dy/dx=y-x for several initial conditions.
Question (JP): dy/dx=y-x の解曲線を複数の初期条件で描け。
Hint: Use solve_ivp.
[Chapter 9 - Problem 18]
Question (EN): Implement code to approximate ∫ f(x,y)dx for system dx/dt=1, dy/dt=y.
Question (JP): 系 dx/dt=1, dy/dt=y に対し ∫ f(x,y)dx を近似せよ。
Hint: Solve y=e^t.
[Chapter 9 - Problem 19]
Question (EN): Solve numerically dy/dx = cos(x)+y with y(0)=0.
Question (JP): dy/dx=cos(x)+y, y(0)=0 を数値的に解け。
Hint: Linear ODE.
[Chapter 9 - Problem 20]
Question (EN): Write Python code to solve Lotka-Volterra predator-prey system numerically.
Question (JP): ロトカ＝ヴォルテラ捕食被食モデルを数値的に解け。
Hint: dx/dt=αx-βxy, dy/dt=δxy-γy.
[Chapter 1 - Problem 1]
Question (EN): Write Python code to represent vector v=(1,2,3) and compute its length.
Question (JP): ベクトル v=(1,2,3) を定義し、その長さを計算せよ。
Hint: Use numpy.linalg.norm.
[Chapter 1 - Problem 2]
Question (EN): Implement vector addition for u=(1,0,1), v=(0,1,1).
Question (JP): ベクトル u=(1,0,1), v=(0,1,1) の加法を実装せよ。
Hint: u+v.
[Chapter 1 - Problem 3]
Question (EN): Compute dot product of u=(1,2), v=(3,4).
Question (JP): ベクトル u=(1,2), v=(3,4) の内積を計算せよ。
Hint: np.dot.
[Chapter 1 - Problem 4]
Question (EN): Compute cross product of u=(1,0,0), v=(0,1,0).
Question (JP): u=(1,0,0), v=(0,1,0) の外積を求めよ。
Hint: Result=(0,0,1).
[Chapter 1 - Problem 5]
Question (EN): Plot vector u=(2,1) in 2D plane.
Question (JP): 2次元平面にベクトル u=(2,1) を描け。
Hint: Use matplotlib.quiver.
[Chapter 1 - Problem 6]
Question (EN): Normalize vector v=(3,4).
Question (JP): ベクトル v=(3,4) を正規化せよ。
Hint: Divide by its norm.
[Chapter 1 - Problem 7]
Question (EN): Compute angle between u=(1,0), v=(0,1).
Question (JP): ベクトル u=(1,0), v=(0,1) のなす角を計算せよ。
Hint: arccos(dot/(‖u‖‖v‖)).
[Chapter 1 - Problem 8]
Question (EN): Represent complex number z=1+i as vector in R^2.
Question (JP): 複素数 z=1+i を R^2 のベクトルとして表せ。
Hint: (1,1).
[Chapter 1 - Problem 9]
Question (EN): Plot unit circle in complex plane.
Question (JP): 複素平面上の単位円を描け。
Hint: Use cosθ,sinθ.
[Chapter 1 - Problem 10]
Question (EN): Implement addition of complex vectors z1=1+2i, z2=2+i.
Question (JP): 複素ベクトル z1=1+2i, z2=2+i の加法を実装せよ。
Hint: Use numpy.complex128.
[Chapter 1 - Problem 11]
Question (EN): Verify distributive law u·(v+w)=u·v+u·w with random vectors.
Question (JP): 任意のベクトルで分配法則 u·(v+w)=u·v+u·w を確認せよ。
Hint: Use numpy.random.
[Chapter 1 - Problem 12]
Question (EN): Implement scalar multiplication of vector v=(1,2,3) with c=3.
Question (JP): ベクトル v=(1,2,3) にスカラー c=3 を掛けよ。
Hint: c*v.
[Chapter 1 - Problem 13]
Question (EN): Compute projection of u=(2,3) on v=(1,0).
Question (JP): u=(2,3) を v=(1,0) 上に射影せよ。
Hint: (u·v/‖v‖^2)v.
[Chapter 1 - Problem 14]
Question (EN): Verify parallelogram law ‖u+v‖^2+‖u-v‖^2=2(‖u‖^2+‖v‖^2).
Question (JP): 平行四辺形の法則を任意のベクトルで確認せよ。
Hint: Use random vectors.
[Chapter 1 - Problem 15]
Question (EN): Write Python code to check linear dependence of {u=(1,2), v=(2,4)}.
Question (JP): ベクトル u=(1,2), v=(2,4) が1次従属か確認せよ。
Hint: Compute determinant=0.
[Chapter 2 - Problem 1]
Question (EN): Define 2×2 matrix A=[[1,2],[3,4]] and print it.
Question (JP): 行列 A=[[1,2],[3,4]] を定義し、出力せよ。
Hint: Use numpy.array.
[Chapter 2 - Problem 2]
Question (EN): Implement addition of A=[[1,2],[3,4]] and B=[[5,6],[7,8]].
Question (JP): A=[[1,2],[3,4]], B=[[5,6],[7,8]] の加法を実装せよ。
Hint: A+B.
[Chapter 2 - Problem 3]
Question (EN): Compute scalar multiplication 3A for A=[[1,2],[3,4]].
Question (JP): A=[[1,2],[3,4]] にスカラー3を掛けよ。
Hint: 3A.
[Chapter 2 - Problem 4]
Question (EN): Compute transpose of A=[[1,2,3],[4,5,6]].
Question (JP): A=[[1,2,3],[4,5,6]] の転置を計算せよ。
Hint: A.T.
[Chapter 2 - Problem 5]
Question (EN): Compute matrix product A=[[1,2],[3,4]], B=[[2,0],[1,2]].
Question (JP): A=[[1,2],[3,4]], B=[[2,0],[1,2]] の積を計算せよ。
Hint: np.dot(A,B).
[Chapter 2 - Problem 6]
Question (EN): Verify associativity (AB)C=A(BC) with random matrices.
Question (JP): 任意の行列で結合法則 (AB)C=A(BC) を確認せよ。
Hint: Use numpy.random.
[Chapter 2 - Problem 7]
Question (EN): Verify distributive law A(B+C)=AB+AC.
Question (JP): 任意の行列で分配法則 A(B+C)=AB+AC を確認せよ。
Hint: Compare results.
[Chapter 2 - Problem 8]
Question (EN): Compute determinant of A=[[1,2],[3,4]].
Question (JP): A=[[1,2],[3,4]] の行列式を求めよ。
Hint: np.linalg.det.
[Chapter 2 - Problem 9]
Question (EN): Compute inverse of A=[[1,2],[3,4]] if it exists.
Question (JP): A=[[1,2],[3,4]] の逆行列を求めよ（存在すれば）。
Hint: np.linalg.inv.
[Chapter 2 - Problem 10]
Question (EN): Verify AA^{-1}=I for A=[[2,1],[7,4]].
Question (JP): A=[[2,1],[7,4]] に対し AA^{-1}=I を確認せよ。
Hint: Use np.allclose.
[Chapter 2 - Problem 11]
Question (EN): Partition matrix A=[[1,2,3,4],[5,6,7,8]] into blocks.
Question (JP): A=[[1,2,3,4],[5,6,7,8]] をブロックに分割せよ。
Hint: Slice numpy arrays.
[Chapter 2 - Problem 12]
Question (EN): Define complex matrix A=[[1+2j,0],[0,3-4j]] and print.
Question (JP): 複素行列 A=[[1+2j,0],[0,3-4j]] を定義せよ。
Hint: Use dtype=complex.
[Chapter 2 - Problem 13]
Question (EN): Compute conjugate transpose of A=[[1+2j,3],[4j,5]].
Question (JP): A=[[1+2j,3],[4j,5]] の随伴行列を求めよ。
Hint: A.conj().T.
[Chapter 2 - Problem 14]
Question (EN): Check if A=[[1,0],[0,1]] is identity.
Question (JP): A=[[1,0],[0,1]] が単位行列か確認せよ。
Hint: Compare with np.eye.
[Chapter 2 - Problem 15]
Question (EN): Write Python code to generate random 3×3 matrix and compute determinant.
Question (JP): ランダムな3×3行列を生成し、その行列式を求めよ。
Hint: np.random.randint.
[Chapter 2 - Problem 16]
Question (EN): Implement matrix multiplication function without numpy.dot.
Question (JP): numpy.dot を使わずに行列積を計算する関数を実装せよ。
Hint: Use loops.
[Chapter 2 - Problem 17]
Question (EN): Compute trace of A=[[1,2],[3,4]].
Question (JP): A=[[1,2],[3,4]] のトレースを求めよ。
Hint: np.trace.
[Chapter 2 - Problem 18]
Question (EN): Verify det(AB)=det(A)det(B) for random 2×2 matrices.
Question (JP): 任意の2×2行列で det(AB)=det(A)det(B) を確認せよ。
Hint: Use np.linalg.det.
[Chapter 2 - Problem 19]
Question (EN): Compute rank of A=[[1,2],[2,4]].
Question (JP): A=[[1,2],[2,4]] のランクを求めよ。
Hint: np.linalg.matrix_rank.
[Chapter 2 - Problem 20]
Question (EN): Write Python code to compute Frobenius norm of A=[[1,2],[3,4]].
Question (JP): A=[[1,2],[3,4]] のフロベニウスノルムを計算せよ。
Hint: np.linalg.norm(A,'fro').
[Chapter 3 - Problem 1]
Question (EN): Define a mapping f: R^2 → R^2 by f(x,y)=(x+y, x−y). Compute f(1,2).
Question (JP): 写像 f: R^2 → R^2, f(x,y)=(x+y, x−y) を定義し、f(1,2) を計算せよ。
Hint: Use simple function.
[Chapter 3 - Problem 2]
Question (EN): Represent f(x,y)=(x+y, x−y) as a matrix and verify.
Question (JP): f(x,y)=(x+y, x−y) を行列で表し、確認せよ。
Hint: [[1,1],[1,-1]].
[Chapter 3 - Problem 3]
Question (EN): Check if T: R^2→R^2 defined by T(x,y)=(x^2,y) is linear.
Question (JP): T(x,y)=(x^2,y) が線形か確認せよ。
Hint: Fails additivity.
[Chapter 3 - Problem 4]
Question (EN): Verify linearity of T(x,y)=(2x,3y).
Question (JP): T(x,y)=(2x,3y) が線形写像であることを確認せよ。
Hint: Check additivity & homogeneity.
[Chapter 3 - Problem 5]
Question (EN): Write Python code to compute matrix of T(x,y)=(x+2y,3x+y).
Question (JP): T(x,y)=(x+2y,3x+y) の行列を求めよ。
Hint: [[1,2],[3,1]].
[Chapter 3 - Problem 6]
Question (EN): Compute composition T2∘T1 where T1(x,y)=(y,x), T2(x,y)=(x,−y).
Question (JP): T1(x,y)=(y,x), T2(x,y)=(x,−y) の合成 T2∘T1 を求めよ。
Hint: Swap then flip sign.
[Chapter 3 - Problem 7]
Question (EN): Represent T1, T2 as matrices and verify matrix product = composition.
Question (JP): T1, T2 を行列で表し、積が合成に一致することを確認せよ。
Hint: Multiply matrices.
[Chapter 3 - Problem 8]
Question (EN): Solve linear system Ax=b with A=[[2,1],[1,3]], b=[1,2].
Question (JP): A=[[2,1],[1,3]], b=[1,2] の連立一次方程式を解け。
Hint: np.linalg.solve.
[Chapter 3 - Problem 9]
Question (EN): Verify T: R^3→R^2, T(x,y,z)=(x+y,z) is linear.
Question (JP): T(x,y,z)=(x+y,z) が線形であることを確認せよ。
Hint: Matrix [[1,1,0],[0,0,1]].
[Chapter 3 - Problem 10]
Question (EN): Write Python function to apply linear map given by matrix A to vector v.
Question (JP): 行列 A で与えられる線形写像をベクトル v に適用する関数を作成せよ。
Hint: return A@v.
[Chapter 3 - Problem 11]
Question (EN): Compute null space of A=[[1,2],[2,4]].
Question (JP): A=[[1,2],[2,4]] の核（零空間）を求めよ。
Hint: Solve Ax=0.
[Chapter 3 - Problem 12]
Question (EN): Compute image of A=[[1,0],[0,0]].
Question (JP): A=[[1,0],[0,0]] の像（値域）を求めよ。
Hint: Span{(1,0)}.
[Chapter 3 - Problem 13]
Question (EN): Verify that composition of two linear maps is linear.
Question (JP): 線形写像の合成が線形であることを確認せよ。
Hint: Use symbolic matrices.
[Chapter 3 - Problem 14]
Question (EN): Show that identity map corresponds to identity matrix.
Question (JP): 恒等写像が単位行列に対応することを示せ。
Hint: I·v=v.
[Chapter 3 - Problem 15]
Question (EN): Write Python code to check if given mapping is linear by testing additivity and homogeneity.
Question (JP): 任意の写像が線形かどうかを、加法性とスカラー倍を確認して判定する関数を書け。
Hint: Use random vectors and scalars.
[Chapter 4 - Problem 1]
Question (EN): Compute determinant of A=[[1,2],[3,4]].
Question (JP): A=[[1,2],[3,4]] の行列式を計算せよ。
Hint: np.linalg.det.
[Chapter 4 - Problem 2]
Question (EN): Verify det(I)=1 for identity matrix of size 3.
Question (JP): 3次の単位行列 I に対して det(I)=1 を確認せよ。
Hint: np.eye.
[Chapter 4 - Problem 3]
Question (EN): Compute determinant of upper triangular matrix A=[[1,2,3],[0,4,5],[0,0,6]].
Question (JP): 上三角行列 A=[[1,2,3],[0,4,5],[0,0,6]] の行列式を計算せよ。
Hint: Product of diagonal entries.
[Chapter 4 - Problem 4]
Question (EN): Implement function to compute 2×2 determinant manually.
Question (JP): 2×2 行列の行列式を手計算で求める関数を実装せよ。
Hint: ad-bc.
[Chapter 4 - Problem 5]
Question (EN): Verify det(AB)=det(A)det(B) with random 3×3 matrices.
Question (JP): 任意の3×3行列で det(AB)=det(A)det(B) を確認せよ。
Hint: np.linalg.det.
[Chapter 4 - Problem 6]
Question (EN): Verify det(A^T)=det(A) with random 3×3 matrix.
Question (JP): 任意の3×3行列で det(A^T)=det(A) を確認せよ。
Hint: np.transpose.
[Chapter 4 - Problem 7]
Question (EN): Show det(cA)=c^n det(A) for scalar c and n×n matrix A.
Question (JP): スカラー c, n×n 行列 A に対し det(cA)=c^n det(A) を確認せよ。
Hint: Use random matrices.
[Chapter 4 - Problem 8]
Question (EN): Write code to compute determinant using Laplace expansion for 3×3 matrix.
Question (JP): 3×3 行列の行列式を余因子展開で求める関数を書け。
Hint: Recursive definition.
[Chapter 4 - Problem 9]
Question (EN): Generate random permutation of [1,2,3] and compute its sign.
Question (JP): [1,2,3] の順列をランダムに生成し、その符号を求めよ。
Hint: Count inversions.
[Chapter 4 - Problem 10]
Question (EN): Verify that swapping two rows changes sign of determinant.
Question (JP): 行を2つ入れ替えると行列式の符号が変わることを確認せよ。
Hint: Compare det before/after.
[Chapter 4 - Problem 11]
Question (EN): Show that if two rows are equal, determinant=0.
Question (JP): 2つの行が等しい場合、行列式が0になることを確認せよ。
Hint: Construct example.
[Chapter 4 - Problem 12]
Question (EN): Compute Vandermonde determinant for [1,2,3].
Question (JP): 値 [1,2,3] に対するヴァンデルモンド行列式を計算せよ。
Hint: det=[[1,1,1],[1,2,4],[1,3,9]].
[Chapter 4 - Problem 13]
Question (EN): Write Python code to test multilinearity of determinant in rows.
Question (JP): 行列式が行に関して多重線形であることを確認せよ。
Hint: Check linearity on row replacement.
[Chapter 4 - Problem 14]
Question (EN): Compute determinant of A=[[0,1],[−1,0]] and interpret geometrically.
Question (JP): A=[[0,1],[−1,0]] の行列式を計算し、幾何的意味を述べよ。
Hint: det=1 (rotation).
[Chapter 4 - Problem 15]
Question (EN): Verify det(diagonal matrix)=product of diagonal entries.
Question (JP): 対角行列の行列式が対角成分の積になることを確認せよ。
Hint: np.diag.
[Chapter 4 - Problem 16]
Question (EN): Compute determinant of random 4×4 matrix using numpy.
Question (JP): ランダムな4×4行列の行列式を求めよ。
Hint: np.random.randn.
[Chapter 4 - Problem 17]
Question (EN): Write function to compute determinant via row-reduction.
Question (JP): 行基本変形を用いて行列式を計算する関数を書け。
Hint: Gaussian elimination.
[Chapter 4 - Problem 18]
Question (EN): Check if det(A)=0 implies matrix is singular.
Question (JP): det(A)=0 なら A が特異行列であることを確認せよ。
Hint: np.linalg.matrix_rank.
[Chapter 4 - Problem 19]
Question (EN): Show that det(A)=det(B) if A is obtained from B by adding multiple of one row to another.
Question (JP): 行列 B のある行に他の行の倍数を加えて得られる行列 A に対し det(A)=det(B) であることを確認せよ。
Hint: Row operation property.
[Chapter 4 - Problem 20]
Question (EN): Compute determinant of Hilbert matrix H_3 with entries 1/(i+j−1).
Question (JP): ヒルベルト行列 H_3（成分 1/(i+j−1)）の行列式を計算せよ。
Hint: Use scipy.linalg.hilbert.
[Chapter 5 - Problem 1]
Question (EN): Solve Ax=b with A=[[2,1],[1,3]], b=[1,2].
Question (JP): A=[[2,1],[1,3]], b=[1,2] の連立一次方程式を解け。
Hint: Use np.linalg.solve.
[Chapter 5 - Problem 2]
Question (EN): Verify solution of Ax=b by computing A·x.
Question (JP): 解 x を代入して A·x が b になることを確認せよ。
Hint: np.allclose.
[Chapter 5 - Problem 3]
Question (EN): Solve 3×3 system Ax=b with A=[[1,1,1],[0,2,5],[2,5,-1]], b=[6,-4,27].
Question (JP): A=[[1,1,1],[0,2,5],[2,5,-1]], b=[6,-4,27] の連立方程式を解け。
Hint: np.linalg.solve.
[Chapter 5 - Problem 4]
Question (EN): Implement Gaussian elimination for solving linear systems.
Question (JP): ガウス消去法を実装して連立一次方程式を解け。
Hint: Forward elimination & back substitution.
[Chapter 5 - Problem 5]
Question (EN): Solve underdetermined system A=[[1,2,3],[2,4,6]], b=[1,2].
Question (JP): A=[[1,2,3],[2,4,6]], b=[1,2] の連立方程式を解け。
Hint: Infinite solutions, use np.linalg.lstsq.
[Chapter 5 - Problem 6]
Question (EN): Solve overdetermined system A=[[1,1],[1,2],[1,3]], b=[1,2,2].
Question (JP): A=[[1,1],[1,2],[1,3]], b=[1,2,2] の連立方程式を解け。
Hint: Least squares solution.
[Chapter 5 - Problem 7]
Question (EN): Implement Cramer’s rule for 2×2 system.
Question (JP): 2×2 の連立方程式をクラーメルの公式で解け。
Hint: x=det(A_x)/det(A).
[Chapter 5 - Problem 8]
Question (EN): Compare performance of Gaussian elimination vs np.linalg.solve for random 100×100 system.
Question (JP): ランダム100×100連立方程式でガウス消去法と np.linalg.solve の性能を比較せよ。
Hint: Use time module.
[Chapter 5 - Problem 9]
Question (EN): Verify uniqueness of solution if det(A)≠0.
Question (JP): det(A)≠0 なら解が一意であることを確認せよ。
Hint: Compute determinant.
[Chapter 5 - Problem 10]
Question (EN): Solve tridiagonal system using specialized solver.
Question (JP): 三重対角行列の連立方程式を専用ソルバで解け。
Hint: Use scipy.linalg.solve_banded.
[Chapter 6 - Problem 1]
Question (EN): Define vector space R^3 and verify closure under addition.
Question (JP): ベクトル空間 R^3 を定義し，加法に関して閉じていることを確認せよ。
Hint: Add two random vectors.
[Chapter 6 - Problem 2]
Question (EN): Verify closure under scalar multiplication in R^3.
Question (JP): R^3 がスカラー倍に関して閉じていることを確認せよ。
Hint: Multiply random vector by scalar.
[Chapter 6 - Problem 3]
Question (EN): Implement Python function to check if given set is subspace.
Question (JP): 与えられた集合が部分空間かを判定する関数を書け。
Hint: Check closure & zero vector.
[Chapter 6 - Problem 4]
Question (EN): Verify that span{(1,0),(0,1)}=R^2.
Question (JP): span{(1,0),(0,1)}=R^2 であることを確認せよ。
Hint: Any (x,y) is linear combination.
[Chapter 6 - Problem 5]
Question (EN): Check if v=(1,2,3) belongs to span{(1,0,0),(0,1,1)}.
Question (JP): v=(1,2,3) が span{(1,0,0),(0,1,1)} に属するか確認せよ。
Hint: Solve linear system.
[Chapter 6 - Problem 6]
Question (EN): Write Python code to compute linear combination a(1,0)+b(0,1).
Question (JP): a(1,0)+b(0,1) の形で線形結合を計算せよ。
Hint: Use random coefficients.
[Chapter 6 - Problem 7]
Question (EN): Verify linear independence of {(1,0,0),(0,1,0),(0,0,1)}.
Question (JP): {(1,0,0),(0,1,0),(0,0,1)} が1次独立であることを確認せよ。
Hint: Determinant ≠ 0.
[Chapter 6 - Problem 8]
Question (EN): Check if {(1,2,3),(2,4,6)} are linearly dependent.
Question (JP): {(1,2,3),(2,4,6)} が1次従属であることを確認せよ。
Hint: One is multiple of other.
[Chapter 6 - Problem 9]
Question (EN): Compute null space of A=[[1,2,3],[4,5,6]].
Question (JP): A=[[1,2,3],[4,5,6]] の零空間を求めよ。
Hint: Solve Ax=0.
[Chapter 6 - Problem 10]
Question (EN): Compute column space of A=[[1,0],[0,1],[1,1]].
Question (JP): A=[[1,0],[0,1],[1,1]] の列空間を求めよ。
Hint: Span of columns.
[Chapter 6 - Problem 11]
Question (EN): Write code to compute basis of span{(1,2),(2,4),(3,6)}.
Question (JP): span{(1,2),(2,4),(3,6)} の基底を求めよ。
Hint: Rank=1.
[Chapter 6 - Problem 12]
Question (EN): Compute dimension of span{(1,0,0),(0,1,0),(0,0,1)}.
Question (JP): span{(1,0,0),(0,1,0),(0,0,1)} の次元を求めよ。
Hint: dim=3.
[Chapter 6 - Problem 13]
Question (EN): Verify that dimension of R^n is n.
Question (JP): R^n の次元が n であることを確認せよ。
Hint: Basis has n vectors.
[Chapter 6 - Problem 14]
Question (EN): Change basis from standard basis to basis {(1,1),(1,−1)} in R^2.
Question (JP): R^2 において標準基底から {(1,1),(1,−1)} への基底変換を行え。
Hint: Construct transition matrix.
[Chapter 6 - Problem 15]
Question (EN): Verify isomorphism between R^2 and span{(1,0),(0,1)}.
Question (JP): R^2 と span{(1,0),(0,1)} の同型を確認せよ。
Hint: Identity mapping.
[Chapter 6 - Problem 16]
Question (EN): Compute coordinates of v=(3,1) relative to basis {(1,1),(1,−1)}.
Question (JP): v=(3,1) を基底 {(1,1),(1,−1)} に関して座標表示せよ。
Hint: Solve linear system.
[Chapter 6 - Problem 17]
Question (EN): Verify that dim(null space)+dim(column space)=n.
Question (JP): 零空間の次元+列空間の次元=n となることを確認せよ。
Hint: Rank-nullity theorem.
[Chapter 6 - Problem 18]
Question (EN): Compute kernel of T:R^2→R^2, T(x,y)=(x+y,x+y).
Question (JP): T(x,y)=(x+y,x+y) の核を求めよ。
Hint: Solve T(x,y)=0.
[Chapter 6 - Problem 19]
Question (EN): Compute image of T(x,y)=(x,0).
Question (JP): T(x,y)=(x,0) の像を求めよ。
Hint: Span{(1,0)}.
[Chapter 6 - Problem 20]
Question (EN): Show that quotient space R^2/span{(1,1)} has dimension 1.
Question (JP): 商空間 R^2/span{(1,1)} の次元が1であることを示せ。
Hint: Dim formula.
[Chapter 6 - Problem 21]
Question (EN): Write Python code to test if set of vectors forms a basis.
Question (JP): 与えられたベクトル集合が基底をなすかを判定せよ。
Hint: Check rank.
[Chapter 6 - Problem 22]
Question (EN): Implement Gram-Schmidt orthogonalization for {(1,1,0),(1,0,1)}.
Question (JP): {(1,1,0),(1,0,1)} に対してグラム・シュミットの正規直交化を行え。
Hint: Use numpy.
[Chapter 6 - Problem 23]
Question (EN): Compute orthogonal complement of span{(1,2)} in R^2.
Question (JP): R^2 における span{(1,2)} の直交補空間を求めよ。
Hint: Solve (x,y)·(1,2)=0.
[Chapter 6 - Problem 24]
Question (EN): Verify that two vector spaces are isomorphic if they have same dimension.
Question (JP): 次元が等しいベクトル空間は同型であることを確認せよ。
Hint: Construct explicit bijection.
[Chapter 6 - Problem 25]
Question (EN): Write Python code to compute basis and dimension of solution space of homogeneous system Ax=0.
Question (JP): 斉次方程式 Ax=0 の解空間の基底と次元を求めよ。
Hint: Use sympy.Matrix.nullspace.
[Chapter 7 - Problem 1]
Question (EN): Compute rank of A=[[1,2],[3,4]].
Question (JP): A=[[1,2],[3,4]] のランクを求めよ。
Hint: np.linalg.matrix_rank.
[Chapter 7 - Problem 2]
Question (EN): Compute rank of A=[[1,2],[2,4]] and explain why rank=1.
Question (JP): A=[[1,2],[2,4]] のランクを求め，なぜランク=1か説明せよ。
Hint: Rows are multiples.
[Chapter 7 - Problem 3]
Question (EN): Write Python code to compute rank of random 5×5 matrix.
Question (JP): ランダムな5×5行列のランクを計算せよ。
Hint: np.random.randint.
[Chapter 7 - Problem 4]
Question (EN): Verify that rank(A)=rank(A^T).
Question (JP): 任意の行列で rank(A)=rank(A^T) を確認せよ。
Hint: Use numpy.
[Chapter 7 - Problem 5]
Question (EN): Check if matrix A=[[1,0,0],[0,0,0],[0,0,1]] is full rank.
Question (JP): A=[[1,0,0],[0,0,0],[0,0,1]] がフルランクか確認せよ。
Hint: Rank=2 < 3.
[Chapter 7 - Problem 6]
Question (EN): Compute rank of product AB with A=[[1,2],[3,4]], B=[[1,0],[0,0]].
Question (JP): A=[[1,2],[3,4]], B=[[1,0],[0,0]] の積 AB のランクを求めよ。
Hint: Compare with rank(A), rank(B).
[Chapter 7 - Problem 7]
Question (EN): Verify rank-nullity theorem for A=[[1,2,3],[4,5,6]].
Question (JP): A=[[1,2,3],[4,5,6]] に対してランク・ヌルリティ定理を確認せよ。
Hint: dim(null)+rank=3.
[Chapter 7 - Problem 8]
Question (EN): Compute rank using minors for A=[[1,2,3],[4,5,6],[7,8,9]].
Question (JP): A=[[1,2,3],[4,5,6],[7,8,9]] の小行列式を用いてランクを求めよ。
Hint: Largest nonzero minor.
[Chapter 7 - Problem 9]
Question (EN): Write function to compute rank via row reduction.
Question (JP): 行基本変形を用いてランクを計算する関数を書け。
Hint: Gaussian elimination.
[Chapter 7 - Problem 10]
Question (EN): Verify that invertible matrix has full rank.
Question (JP): 正則行列がフルランクであることを確認せよ。
Hint: det≠0 ⇒ rank=n.
[Chapter 8 - Problem 1]
Question (EN): Solve Ax=b with A=[[1,2,1],[0,1,2],[2,3,4]], b=[4,5,6].
Question (JP): A=[[1,2,1],[0,1,2],[2,3,4]], b=[4,5,6] の連立一次方程式を解け。
Hint: np.linalg.solve.
[Chapter 8 - Problem 2]
Question (EN): Check consistency of system A=[[1,2],[2,4]], b=[1,3].
Question (JP): A=[[1,2],[2,4]], b=[1,3] の連立方程式が整合的か確認せよ。
Hint: Use rank(A)=rank([A|b]).
[Chapter 8 - Problem 3]
Question (EN): Implement Gaussian elimination with partial pivoting.
Question (JP): 部分ピボット選択付きガウス消去法を実装せよ。
Hint: Swap rows when pivot=0.
[Chapter 8 - Problem 4]
Question (EN): Write Python code to transform matrix A=[[2,1],[4,2]] into row echelon form.
Question (JP): 行列 A=[[2,1],[4,2]] を行階段形に変形せよ。
Hint: Use row operations.
[Chapter 8 - Problem 5]
Question (EN): Solve homogeneous system Ax=0 with A=[[1,2,3],[2,4,6]].
Question (JP): A=[[1,2,3],[2,4,6]] に対する斉次連立方程式 Ax=0 を解け。
Hint: Null space basis.
[Chapter 8 - Problem 6]
Question (EN): Solve system using LU decomposition for A=[[3,1],[1,2]], b=[9,8].
Question (JP): A=[[3,1],[1,2]], b=[9,8] をLU分解で解け。
Hint: Use scipy.linalg.lu.
[Chapter 8 - Problem 7]
Question (EN): Compute inverse of A=[[1,2],[3,4]] using Gaussian elimination.
Question (JP): ガウス消去法で A=[[1,2],[3,4]] の逆行列を計算せよ。
Hint: Augment with identity.
[Chapter 8 - Problem 8]
Question (EN): Write Python function to perform back substitution for upper triangular system.
Question (JP): 上三角行列の連立方程式を解く後退代入の関数を実装せよ。
Hint: Solve last row first.
[Chapter 8 - Problem 9]
Question (EN): Solve Ax=b with A=[[1,1,1],[0,1,2],[0,0,1]], b=[6,5,3] by back substitution.
Question (JP): A=[[1,1,1],[0,1,2],[0,0,1]], b=[6,5,3] を後退代入で解け。
Hint: Triangular system.
[Chapter 8 - Problem 10]
Question (EN): Implement forward substitution for lower triangular system.
Question (JP): 下三角行列の連立一次方程式を解く前進代入を実装せよ。
Hint: Start from top row.
[Chapter 8 - Problem 11]
Question (EN): Verify uniqueness of solution when rank(A)=n.
Question (JP): rank(A)=n のとき解が一意であることを確認せよ。
Hint: Square invertible matrix.
[Chapter 8 - Problem 12]
Question (EN): Implement iterative Jacobi method for Ax=b.
Question (JP): Jacobi 法を実装して Ax=b を解け。
Hint: Update each variable using previous iteration.
[Chapter 8 - Problem 13]
Question (EN): Implement Gauss-Seidel method for Ax=b.
Question (JP): Gauss-Seidel 法を実装して Ax=b を解け。
Hint: Update sequentially.
[Chapter 8 - Problem 14]
Question (EN): Compare convergence speed of Jacobi vs Gauss-Seidel on same system.
Question (JP): Jacobi 法と Gauss-Seidel 法の収束速度を比較せよ。
Hint: Plot errors per iteration.
[Chapter 8 - Problem 15]
Question (EN): Solve band matrix system efficiently using scipy.
Question (JP): バンド行列の連立一次方程式を scipy で効率的に解け。
Hint: Use solve_banded.
[Chapter 8 - Problem 16]
Question (EN): Solve sparse system using scipy.sparse.linalg.spsolve.
Question (JP): 疎行列の連立一次方程式を spsolve で解け。
Hint: Generate sparse matrix.
[Chapter 8 - Problem 17]
Question (EN): Verify that row operations do not change solution set.
Question (JP): 行基本変形が解集合を変えないことを確認せよ。
Hint: Transform A and b together.
[Chapter 8 - Problem 18]
Question (EN): Write code to compute reduced row echelon form (RREF).
Question (JP): 行列の簡約階段形 (RREF) を求める関数を書け。
Hint: Normalize pivot rows.
[Chapter 8 - Problem 19]
Question (EN): Solve Ax=b with A random 10×10 and b random 10-vector.
Question (JP): ランダム10×10行列 A とベクトル b に対して Ax=b を解け。
Hint: np.linalg.solve.
[Chapter 8 - Problem 20]
Question (EN): Verify equivalence between invertibility of A and uniqueness of solution Ax=b.
Question (JP): 行列 A が可逆であることと Ax=b の解が一意であることが同値であることを確認せよ。
Hint: det(A)≠0 ⇔ unique solution.
[Chapter 9 - Problem 1]
Question (EN): Compute eigenvalues of A=[[2,0],[0,3]].
Question (JP): A=[[2,0],[0,3]] の固有値を求めよ。
Hint: Diagonal entries.
[Chapter 9 - Problem 2]
Question (EN): Find eigenvectors of A=[[2,0],[0,3]].
Question (JP): A=[[2,0],[0,3]] の固有ベクトルを求めよ。
Hint: Standard basis.
[Chapter 9 - Problem 3]
Question (EN): Compute eigenvalues of A=[[1,1],[0,1]].
Question (JP): A=[[1,1],[0,1]] の固有値を求めよ。
Hint: Characteristic polynomial.
[Chapter 9 - Problem 4]
Question (EN): Compute eigenvalues of A=[[0,-1],[1,0]].
Question (JP): A=[[0,-1],[1,0]] の固有値を求めよ。
Hint: Complex roots.
[Chapter 9 - Problem 5]
Question (EN): Diagonalize A=[[4,1],[0,4]] if possible.
Question (JP): A=[[4,1],[0,4]] を対角化せよ（可能なら）。
Hint: Check multiplicity vs dimension.
[Chapter 9 - Problem 6]
Question (EN): Compute eigenvalues of A=[[2,1],[1,2]].
Question (JP): A=[[2,1],[1,2]] の固有値を求めよ。
Hint: Solve det(A-λI)=0.
[Chapter 9 - Problem 7]
Question (EN): Find eigenvectors of A=[[2,1],[1,2]].
Question (JP): A=[[2,1],[1,2]] の固有ベクトルを求めよ。
Hint: Symmetric matrix.
[Chapter 9 - Problem 8]
Question (EN): Write Python code to compute eigenvalues of random 3x3 matrix.
Question (JP): ランダム3×3行列の固有値を求めるPythonコードを書け。
Hint: np.linalg.eig.
[Chapter 9 - Problem 9]
Question (EN): Show that determinant equals product of eigenvalues.
Question (JP): 行列式が固有値の積に等しいことを示せ。
Hint: Characteristic polynomial.
[Chapter 9 - Problem 10]
Question (EN): Show that trace equals sum of eigenvalues.
Question (JP): トレースが固有値の和に等しいことを示せ。
Hint: Characteristic polynomial.
[Chapter 9 - Problem 11]
Question (EN): Compute eigenvalues of A=[[0,1],[1,0]].
Question (JP): A=[[0,1],[1,0]] の固有値を求めよ。
Hint: ±1.
[Chapter 9 - Problem 12]
Question (EN): Check diagonalizability of A=[[1,1],[0,1]].
Question (JP): A=[[1,1],[0,1]] が対角化可能か調べよ。
Hint: Jordan form.
[Chapter 9 - Problem 13]
Question (EN): Compute eigenvalues of rotation matrix [[cosθ,-sinθ],[sinθ,cosθ]].
Question (JP): 回転行列 [[cosθ,-sinθ],[sinθ,cosθ]] の固有値を求めよ。
Hint: Complex exponential.
[Chapter 9 - Problem 14]
Question (EN): Compute eigenvalues of scaling matrix [[k,0],[0,m]].
Question (JP): 拡大縮小行列 [[k,0],[0,m]] の固有値を求めよ。
Hint: k,m.
[Chapter 9 - Problem 15]
Question (EN): Show that similar matrices have same eigenvalues.
Question (JP): 相似な行列は固有値が等しいことを示せ。
Hint: det(P⁻¹AP-λI).
[Chapter 9 - Problem 16]
Question (EN): Write code to verify orthogonality of eigenvectors of symmetric matrix.
Question (JP): 対称行列の固有ベクトルが直交することを確認するコードを書け。
Hint: np.allclose(v1·v2,0).
[Chapter 9 - Problem 17]
Question (EN): Compute eigenvalues of A=[[1,2],[2,1]].
Question (JP): A=[[1,2],[2,1]] の固有値を求めよ。
Hint: λ=3,-1.
[Chapter 9 - Problem 18]
Question (EN): Find eigenvectors of A=[[1,2],[2,1]].
Question (JP): A=[[1,2],[2,1]] の固有ベクトルを求めよ。
Hint: Solve (A-λI)x=0.
[Chapter 9 - Problem 19]
Question (EN): Compute characteristic polynomial of A=[[a,b],[c,d]].
Question (JP): A=[[a,b],[c,d]] の特性多項式を求めよ。
Hint: λ²-(a+d)λ+(ad-bc).
[Chapter 9 - Problem 20]
Question (EN): Implement function to compute eigen decomposition of 2x2 real matrix.
Question (JP): 実2×2行列の固有分解を求める関数を実装せよ。
Hint: Return eigenvalues, eigenvectors.
[Chapter 10 - Problem 1]
Question (EN): Define inner product in R^n. Give formula.
Question (JP): R^n における内積を定義し，公式を示せ。
Hint: ⟨x,y⟩=Σ x_i y_i.
[Chapter 10 - Problem 2]
Question (EN): Compute inner product of x=[1,2,3], y=[4,5,6].
Question (JP): x=[1,2,3], y=[4,5,6] の内積を求めよ。
Hint: Dot product.
[Chapter 10 - Problem 3]
Question (EN): Verify Cauchy-Schwarz inequality in R^2 with x=[1,2], y=[3,4].
Question (JP): R^2 の x=[1,2], y=[3,4] でコーシー・シュワルツの不等式を確認せよ。
[Chapter 10 - Problem 4]
Question (EN): Show that ||x||=sqrt(⟨x,x⟩).
Question (JP): ||x||=√⟨x,x⟩ を示せ。
[Chapter 10 - Problem 5]
Question (EN): Compute angle between x=[1,0], y=[0,1].
Question (JP): x=[1,0], y=[0,1] のなす角を求めよ。
Hint: 90°.
[Chapter 10 - Problem 6]
Question (EN): Verify parallelogram law for x=[1,2], y=[3,4].
Question (JP): x=[1,2], y=[3,4] に対して平行四辺形法則を確認せよ。
[Chapter 10 - Problem 7]
Question (EN): Implement function to compute inner product in Python.
Question (JP): Pythonで内積を計算する関数を実装せよ。
Hint: np.dot.
[Chapter 10 - Problem 8]
Question (EN): Perform Gram-Schmidt orthogonalization for v1=[1,1], v2=[1,0].
Question (JP): v1=[1,1], v2=[1,0] に対してグラム・シュミットの正規直交化を行え。
[Chapter 10 - Problem 9]
Question (EN): Normalize vector v=[3,4].
Question (JP): v=[3,4] を正規化せよ。
[Chapter 10 - Problem 10]
Question (EN): Verify orthogonality of u=[1,-1], v=[1,1].
Question (JP): u=[1,-1], v=[1,1] が直交することを確認せよ。
[Chapter 10 - Problem 11]
Question (EN): Compute projection of v=[2,3] onto u=[1,0].
Question (JP): v=[2,3] を u=[1,0] に射影せよ。
[Chapter 10 - Problem 12]
Question (EN): Show projection matrix P=uu^T/(u^Tu).
Question (JP): 射影行列 P=uu^T/(u^Tu) を示せ。
[Chapter 10 - Problem 13]
Question (EN): Compute orthogonal complement of span{[1,2]} in R^2.
Question (JP): R^2 における span{[1,2]} の直交補空間を求めよ。
[Chapter 10 - Problem 14]
Question (EN): Verify ⟨Ax,y⟩=⟨x,A^Ty⟩ for A=[[1,2],[0,1]].
Question (JP): A=[[1,2],[0,1]] について ⟨Ax,y⟩=⟨x,A^Ty⟩ を確認せよ。
[Chapter 10 - Problem 15]
Question (EN): Show that orthogonal matrix preserves inner product.
Question (JP): 直交行列が内積を保つことを示せ。
[Chapter 10 - Problem 16]
Question (EN): Compute determinant of orthogonal matrix A=[[0,1],[-1,0]].
Question (JP): 直交行列 A=[[0,1],[-1,0]] の行列式を求めよ。
Hint: det=±1.
[Chapter 10 - Problem 17]
Question (EN): Verify Pythagoras theorem using inner product for x=[3,0], y=[0,4].
Question (JP): x=[3,0], y=[0,4] で内積を用いてピタゴラスの定理を確認せよ。
[Chapter 10 - Problem 18]
Question (EN): Implement function to compute cosine similarity.
Question (JP): コサイン類似度を計算する関数を実装せよ。
[Chapter 10 - Problem 19]
Question (EN): Define Hermitian inner product on C^n.
Question (JP): C^n 上のエルミート内積を定義せよ。
Hint: ⟨x,y⟩=Σ x_i conj(y_i).
[Chapter 10 - Problem 20]
Question (EN): Verify unitary matrix property U*U=I with U=[[0,1],[-1,0]].
Question (JP): U=[[0,1],[-1,0]] がユニタリ行列であることを確認せよ。
[Chapter 10 - Problem 21]
Question (EN): Compute length of complex vector z=[1+i,2-i].
Question (JP): 複素ベクトル z=[1+i,2-i] の長さを求めよ。
[Chapter 10 - Problem 22]
Question (EN): Verify orthogonality of Fourier basis vectors in C^n.
Question (JP): フーリエ基底ベクトルが C^n で直交することを確認せよ。
[Chapter 10 - Problem 23]
Question (EN): Show that ⟨x,y⟩=0 implies x,y are orthogonal.
Question (JP): ⟨x,y⟩=0 のとき，x と y が直交することを示せ。
[Chapter 10 - Problem 24]
Question (EN): Compute orthogonal projection of v=[2,2,1] onto span{[1,0,0],[0,1,0]}.
Question (JP): v=[2,2,1] を span{[1,0,0],[0,1,0]} に射影せよ。
[Chapter 10 - Problem 25]
Question (EN): Implement QR decomposition using Gram-Schmidt.
Question (JP): グラム・シュミットを用いて QR分解を実装せよ。
[Chapter 11 - Problem 1]
Question (EN): Define a normal matrix. Give condition in terms of A and A*.
Question (JP): 正規行列を定義し，A と A* による条件を示せ。
Hint: AA*=A*A.
[Chapter 11 - Problem 2]
Question (EN): Show that symmetric matrices are normal.
Question (JP): 対称行列は正規行列であることを示せ。
[Chapter 11 - Problem 3]
Question (EN): Show that Hermitian matrices are diagonalizable with real eigenvalues.
Question (JP): エルミート行列が実固有値をもち対角化可能であることを示せ。
[Chapter 11 - Problem 4]
Question (EN): Diagonalize A=[[2,1],[1,2]].
Question (JP): A=[[2,1],[1,2]] を対角化せよ。
Hint: Use orthogonal eigenvectors.
[Chapter 11 - Problem 5]
Question (EN): Show that unitary matrices are normal.
Question (JP): ユニタリ行列が正規行列であることを示せ。
[Chapter 11 - Problem 6]
Question (EN): Compute eigenvalues of Hermitian matrix [[1,i],[-i,1]].
Question (JP): エルミート行列 [[1,i],[-i,1]] の固有値を求めよ。
[Chapter 11 - Problem 7]
Question (EN): Verify orthogonality of eigenvectors of symmetric matrix [[3,2],[2,3]].
Question (JP): 対称行列 [[3,2],[2,3]] の固有ベクトルの直交性を確認せよ。
[Chapter 11 - Problem 8]
Question (EN): Write Python code to check if a given matrix is normal.
Question (JP): 行列が正規行列かどうかを判定するPythonコードを書け。
Hint: np.allclose(A@A.conj().T, A.conj().T@A).
[Chapter 11 - Problem 9]
Question (EN): Diagonalize rotation matrix [[0,-1],[1,0]] over C.
Question (JP): 回転行列 [[0,-1],[1,0]] を C 上で対角化せよ。
[Chapter 11 - Problem 10]
Question (EN): Explain why every normal matrix is unitarily diagonalizable.
Question (JP): すべての正規行列がユニタリ対角化可能である理由を説明せよ。
Hint: Spectral theorem.
[Chapter 12 - Problem 1]
Question (EN): Define Jordan canonical form (JCF).
Question (JP): ジョルダン標準形 (JCF) を定義せよ。
Hint: Block diagonal with Jordan blocks.
[Chapter 12 - Problem 2]
Question (EN): Give Jordan form of A=[[1,1],[0,1]].
Question (JP): A=[[1,1],[0,1]] のジョルダン標準形を求めよ。
Hint: One Jordan block with λ=1.
[Chapter 12 - Problem 3]
Question (EN): Compute Jordan form of A=[[2,0,0],[0,2,1],[0,0,2]].
Question (JP): A=[[2,0,0],[0,2,1],[0,0,2]] のジョルダン標準形を求めよ。
[Chapter 12 - Problem 4]
Question (EN): Explain meaning of generalized eigenvector.
Question (JP): 一般化固有ベクトルの意味を説明せよ。
[Chapter 12 - Problem 5]
Question (EN): Find Jordan form of nilpotent matrix [[0,1,0],[0,0,1],[0,0,0]].
Question (JP): 冪零行列 [[0,1,0],[0,0,1],[0,0,0]] のジョルダン標準形を求めよ。
Hint: Single Jordan block.
[Chapter 12 - Problem 6]
Question (EN): Write Python code to compute Jordan form using sympy.
Question (JP): sympy を使ってジョルダン標準形を求めるコードを書け。
Hint: sympy.Matrix.jordan_form().
[Chapter 12 - Problem 7]
Question (EN): Explain why every square matrix over C has a Jordan form.
Question (JP): C 上の任意の正方行列がジョルダン標準形をもつ理由を説明せよ。
Hint: Fundamental theorem of algebra.
[Chapter 12 - Problem 8]
Question (EN): Compute Jordan form of A=[[5,4],[0,5]].
Question (JP): A=[[5,4],[0,5]] のジョルダン標準形を求めよ。
[Chapter 12 - Problem 9]
Question (EN): Explain difference between diagonalization and Jordan form.
Question (JP): 対角化とジョルダン標準形の違いを説明せよ。
[Chapter 12 - Problem 10]
Question (EN): Find Jordan form of A=[[3,1,0],[0,3,1],[0,0,3]].
Question (JP): A=[[3,1,0],[0,3,1],[0,0,3]] のジョルダン標準形を求めよ。
Hint: One Jordan block of size 3.