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 the variable 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 0 0 2 0 0 0 0 0 2 0 0 0 0 0 1 −1 0 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. Prove that W is a subspace of of V.

(b) [12 points] Let U be the subspace of W of functions f which also satisfy

dn f dxn

(−1) = dn f dxn

(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 linear operator defined by

D( f )(x) = d

dx f (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 not need to prove that B is a basis).

(b) [10 points] Find the matrix of orthogonal projection onto W and hence find the closest vector 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 quadratic form Q(y1, y2, y3) relative to this basis.

(d) [2 points] Classify the quadric surface Q(x1, x2, x3) = −1.