Department of Mathematics
MATHS 253 Assignment 2 due 11 April 2022
This assignment will be marked out of a total of 100. Please submit your answers via Canvas,
in the usual way, by 11:59 pm on the due date.
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 = 1
3
1 0 4
0 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) = x
2 be the standard basis of V. Define the
inner product h·, ·i on V by
hf , gi =
Z 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) = x
4 on [0, 1].
5. [20 points] Let Q be the quadratic form given by
Q(x1, x2, x3) = αx
2
1 + 2x1x2 + αx
2
2 + 4x1x3 + 2x2x3 − 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 number λ0 (which is the same for all the blocks). What are pA(λ) and µA(λ)? Prove that your answer
is correct
