MATHS 253 Semester Test 13 April 2022
This test is marked out of 100.
1. [20 points] Let V = R2[x] be the vector space of all real polynomials of degree at most 2 in thevariable x, and let D : V → V be the linear operator defined by
D( f )(x) = f (x) + (x − 1) f ′(x) + x f ′(x − 1) (1)
(you do not need to prove that D is a linear operator).
(a) [10 points] Let B = {1, x, x2}. Find [D]B, the matrix of D relative to the basis B.(b) [10 points] Prove that D is invertible. Find a polynomial f ∈ R2[x] such that
D( f )(x) = 5×2 + 2x + 1. (2)
2. [20 points] Let
A =
2 0 1 0 00 2 0 0 00 0 2 0 00 0 0 1 −10 0 0 0 1
(a) [5 points] Write down the characteristic polynomial of A. What are the eigenvalues of A?(b) [15 points] For each eigenvalue of A, find its geometric and algebraic multiplicities. Is A
diagonalizable?
3. [20 points] Let V = C∞R[−1, 1], the space of infinitely-differentiable functions f : [−1, 1] → R.
(a) [5 points] Let W be the subset of V of functions f which satisfy f (−1) = f (1) = 0. Provethat W is a subspace of of V.
(b) [12 points] Let U be the subspace of W of functions f which also satisfy
dn fdxn
(−1) =dn fdxn
(1) = 0 for all n ∈ N
(you do not need to prove that U is a subspace of W). Define an inner product on U by
( f , g) =∫ 1−1
f (t)g(t) dt
(you do not need to prove that this is an inner product). Let D : U → U be the linearoperator defined by
D( f )(x) =d
dxf (x).
(you do not need to prove that D is a linear operator). Prove that D satisfies D∗ = −D.Such an operator is called skew-Hermitian.
Hint. Use integration by parts.
(c) [3 points] Using part (b), show that if f ∈ U, then f is orthogonal to its derivative.
QUESTIONS CONTINUE ON NEXT PAGE
4. [20 points] Let W ⊆ R3 be the plane given by x − 2y + z = 0.
(a) [10 points] Starting with the basis
B =
11
1
,
32
1
for W, run the Gram-Schmidt algorithm to find an orthonormal basis for W. (You do notneed to prove that B is a basis).
(b) [10 points] Find the matrix of orthogonal projection onto W and hence find the closestvector v ∈ W to x = (1, 0, 0)T .
5. [20 points] Let
A =
−1 0 00 −3 1
0 1 −3
(a) [2 points] Write down the quadratic form Q(x1, x2, x3) which A represents.
(b) [6 points] Is Q positive definite, negative definite, or indefinite? Justify your answer.
(c) [10 points] Find a basis of R3 in which Q has no cross-terms. Write down the quadraticform Q(y1, y2, y3) relative to this basis.
(d) [2 points] Classify the quadric surface Q(x1, x2, x3) = −1.
