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MATH 235 — Linear Algebra II

Midterm 2 · 75 minutes · Total: 60 marks

Q1 [10 marks] — Eigenvalues

Let A be the 3×3 matrix with rows [2, 1, 0], [0, 3, 1], [0, 0, 2].

(a) Find all eigenvalues of A. [4 marks]

(b) For each eigenvalue, find a basis for the eigenspace. [6 marks]

Q2 [10 marks] — Diagonalization

Let B have eigenvalues λ₁ = 1, λ₂ = 4, λ₃ = 4 with corresponding eigenspaces of dimension 1, 2.

(a) Is B diagonalizable? Justify. [4 marks]

(b) If yes, find an invertible P and diagonal D such that B = PDP⁻¹. [6 marks]

Q3 [10 marks] — Orthogonality

Let W = span{v₁, v₂} where v₁ = [1, 1, 0, 1]ᵀ and v₂ = [0, 1, 1, 0]ᵀ.

(a) Apply Gram-Schmidt to {v₁, v₂} to produce an orthonormal basis for W. [6 marks]

(b) Find the orthogonal projection of b = [3, 1, 2, 4]ᵀ onto W. [4 marks]

Q4 [8 marks] — Least Squares

The system Ax = b has no solution. Find the least-squares solution x̂ where:

A = [[1,1],[1,2],[1,3]]   b = [1, 3, 4]ᵀ

Q5 [12 marks] — Symmetric Matrices

Let S be a real symmetric matrix with eigenvalues 2 and 5.

(a) Prove that the eigenspaces of S are orthogonal. [6 marks]

(b) Find an orthogonal matrix Q and diagonal D such that S = QDQᵀ. [6 marks]

Q6 [10 marks] — Complex Eigenvalues

The matrix C = [[0, −2], [2, 0]] has complex eigenvalues.

(a) Find the eigenvalues and eigenvectors of C. [6 marks]

(b) Describe the geometric effect of C on ℝ². [4 marks]

MATH 235 — Linear Algebra II

Practice Midterm 2 · 75 minutes · Total: 60 marks

Q1 [10 marks] — Eigenvalues & Eigenvectors

Consider the matrix M = [[3, 1, 0], [0, 3, 0], [0, 0, 5]].

(a) Find all eigenvalues of M and their algebraic multiplicities. [4 marks]

(b) Determine a basis for each eigenspace. Is M diagonalizable? [6 marks]

Q2 [10 marks] — Diagonalization

Suppose A is a 3×3 matrix with characteristic polynomial p(λ) = −(λ − 2)²(λ − 7).

(a) Under what conditions on the eigenspaces is A diagonalizable? [4 marks]

(b) Given that A is diagonalizable, compute A⁴. [6 marks]

Q3 [10 marks] — Orthogonal Projections

Let W be the subspace of ℝ⁴ spanned by u₁ = [1, 0, 1, 1]ᵀ and u₂ = [1, 1, 0, −1]ᵀ.

(a) Verify that {u₁, u₂} is orthogonal. [2 marks]

(b) Find projᵣ y for y = [2, 3, 1, 0]ᵀ. [4 marks]

(c) Find the distance from y to W. [4 marks]

Q4 [8 marks] — Least Squares Regression

Fit a line y = c₀ + c₁t to the data points (0, 1), (1, 2), (2, 4) using least squares.

Set up and solve the normal equations AᵀAx̂ = Aᵀb.

Q5 [12 marks] — Spectral Theorem

Let S = [[5, 2], [2, 2]] be a real symmetric matrix.

(a) Find the eigenvalues of S. [4 marks]

(b) Find an orthonormal eigenbasis and write the orthogonal diagonalization S = QDQᵀ. [8 marks]

Q6 [10 marks] — Change of Basis

Let B = {[1,1]ᵀ, [1,−1]ᵀ} and B' = {[2,1]ᵀ, [1,1]ᵀ} be two bases for ℝ².

(a) Find the change-of-basis matrix from B to B'. [6 marks]

(b) If [v]ᵣ = [3, −1]ᵀ, find [v]ᵣ'. [4 marks]