Section 3: Problems

University-level exam questions for Geometric & Optimization Methods.

Orthogonality

Problem 1.1

Verify that the vectors v₁ = (1, 1, 1) and v₂ = (1, -2, 1) are orthogonal.

Difficulty: Easy

Problem 1.2

Find an orthogonal basis for the column space of A = [[1, 1], [1, 2], [1, 3]].

Difficulty: Medium

Problem 1.3

Prove that if $\{q₁, q₂, ..., qₙ\}$ is an orthonormal basis, then any vector v can be written as $v = \sum(v \cdot qᵢ)qᵢ$.

Difficulty: Medium

Projections

Problem 2.1

Find the projection of b = (1, 2, 3) onto the line through a = (1, 1, 1).

Difficulty: Easy

Problem 2.2

Prove that for projection matrix P, we have P² = P.

Difficulty: Medium

Least Squares

Problem 3.1

Fit a line y = C + Dt to data points: (0, 6), (1, 0), (2, 0).

Difficulty: Medium

Problem 3.2

For matrix A with full column rank, prove that AᵀA is invertible.

Difficulty: Hard

Gram-Schmidt

Problem 4.1

Apply Gram-Schmidt to vectors a₁ = (1, 1, 0), a₂ = (1, 0, 1), a₃ = (0, 1, 1).

Difficulty: Medium

Problem 4.2

Find the QR decomposition of A = [[1, 1], [1, 2], [1, 3]].

Difficulty: Medium

Optimization

Problem 5.1

Find the minimum of f(x, y) = x² + 2xy + 3y² - 2x - 4y.

Difficulty: Medium

Problem 5.2

Use Lagrange multipliers to minimize f(x, y) = x² + y² subject to x + y = 1.

Difficulty: Hard

Challenge Problems

Problem 6.1

Prove that the four fundamental subspaces come in orthogonal pairs.

Difficulty: Very Hard

Problem 6.2

Derive the normal equations from first principles using calculus.

Difficulty: Hard

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Solutions

Solutions are available in the implementation file with verification code.