MATHS 253 Assignment - Department

Department of Mathematics

MATHS 253 Assignment 2 due 11 April 2022

This assignment will be marked out of a total of 100.

1. [15 points] Let f : [−1, 1] → R be defined by f (x) = ??? ?? −1 if − 1 ≤ x < 0 0 if x = 0 1 if 0 < x ≤ 1

(a) [3 points] Is f odd, even, neither, or both? (b) [12 points] Find the 6th degree Fourier series approximation to f .

2. [25 points] A real orthogonal matrix A is called special orthogonal if det A = 1.

(a) [5 points] Let A and B be orthogonal matrices. Show that AB is special orthogonal if and only if both A and B are special orthogonal, or neither A nor B is special orthogonal.

(b) [20 points] Let A ∈ R2×2 be special orthogonal. Show that there exists ϑ ∈ [0, 2π) such that

A = [cos ϑ −sin ϑ sin ϑ cos ϑ]

3. [20 points] Let A = 13 ? ? 1 0 40 5 −4

4 −4 3? ?. Compute the spectral decomposition of A and use it to compute sin (π 6 A ) .

4. [20 points] Let V = R2[x] be the real vector space of polynomials of degree at most 2. Let B = { f0, f1, f2} with f0(x) = 1, f1(x) = x, and f2(x) = x2 be the standard basis of V. Define the inner product ?·, ·? on V by

? f , g? = ∫ 1 0 f (x)g(x) dx.

(a) [10 points] Use the Gram-Schmidt algorithm on B to find an orthogonal basis of V. (b) [10 points] Find the best quadratic approximation to f (x) = x4 on [0, 1].

5. [20 points] Let Q be the quadratic form given by

Q(x1, x2, x3) = αx 2 1 + 2x1 x2 + αx

2 2 + 4x1 x3 + 2x2 x3 − 3x

2 3

Find all values of α ∈ R such that Q is positive definite, and all values of α ∈ R such that Q is negative definite, if they exist.

6. [Bonus question for up to 5 points of extra credit, if required] Let A be a Jordan matrix whose Jordan blocks are Jn1(λ0), Jn2(λ0), . . . , Jnk (λ0) for some integers n1, n2, . . . nk and some real num- ber λ0 (which is the same for all the blocks). What are pA(λ) and µA(λ)? Prove that your answer is correct.