M1: Linear Algebra II

Question 1

Theorem

Criterion for Invertibility

Answer 1

Let A be an n x n matrix
det A ≠ 0 iff A is invertible
In that case det A⁻¹ = 1/(det A)

Question 2

Proof

(Criterion for Invertibility)
Let A be an n x n matrix
det A ≠ 0 iff A is invertible
In that case det A⁻¹ = 1/(det A)

2 points

Answer 2

• If A is not invertible then E₁…EₖA = R and det R = 0
• If A is invertible then E₁…EₖA = Iₙ

Question 3

Theorem

Product Rule for Determinants

Answer 3

det AB = det A det B

Question 4

Proof

(Product Rule for Determinants)
det AB = det A det B

2 points

Answer 4

• Consider cases where AB is singular and invertible
• Use product rule for elementary matrices

Question 5

Theorem

Transpose Rule for Determinants

Answer 5

det Aᵀ = det A

Question 6

Proof

(Transpose Rule for Determinants)
det Aᵀ = det A

2 points

Answer 6

• Consider cases where A is singular and invertible
• Use product rule for elementary matrices

Question 7

Theorem

Gram-Schmidt Orthogonalization Process GSOP

Answer 7

Let v̲₁, …, v̲ₖ be k independent vectors. Then there is an orthogonal set w̲₁, …, w̲ₖ such that
<v̲₁, …, v̲ᵢ> = <w̲₁, …, w̲ᵢ> for i = 1, 2, …, k

Question 8

Proof

(Gram-Schmidt Orthogonalization Process GSOP)
Let v̲₁, …, v̲ₖ be k independent vectors. Then there is an orthogonal set w̲₁, …, w̲ₖ such that
<v̲₁, …, v̲ᵢ> = <w̲₁, …, w̲ᵢ> for i = 1, 2, …, k

2 points

Answer 8

• Induct on i
• Consider y̲ᵢ₊₁ = v̲ᵢ₊₁ - (v̲ᵢ₊₁⋅w̲₁)w̲₁ - … - (v̲ᵢ₊₁⋅w̲ᵢ)w̲ᵢ

Question 9

Theorem

The Spectal Theorem

Answer 9

Let A be a n x n symmetric matrix. Then A has an orthonormal eigenbasis

Question 10

Proof

(The Spectral Theorem)
Let A be a n x n symmetric matrix. Then A has an orthonormal eigenbasis

Answer 10

• Strongly induct on n
• Let v̲₁, …, v̲ₖ be an orthonormal basis of the λ-eigenspace
• Extend to an orthonormal basis of ℝⁿ𝒸ₒₗ
• Place as the columns of P
• Apply inductive hypothesis to bottom-right (n - k) x (n - k) matrix of PᵀAP