Practice final, Math 113.

• If you run out of time to write detailed proofs, write an outline of the proof.• If you cannot figure out how to start, try to work out an example; partial

credit will be given for correctly worked examples.• The exam is intentionally long; don’t be discouraged if you do not finish!• Questions marked with ? are harder, and I recommend that you attempt

them at the end.

Problem 1. No justification is needed. In this question, V is an n-dimensional vectorspace over C, T, S are linear operators from V to itself, and W is an m-dimensional inner product space over R.

Mark true or false.(a) Any spanning set for V contains a basis.(b) V is isomorphic to its dual space V ∗.(c) Suppose W1,W2 are two subspaces of W . Then there exists an isom-

etry σ : W →W so that σ(W1) = W2.(d) If TS is zero, then ST is zero.(e) If TS is not injective, then ST is not injective.(f) If TS is the identity operator, then ST is the identity operator.(g) The characteristic polynomials of TS and ST are the same.(h) Any linear transformation from W to itself has an eigenvector.(i) If x, y ∈W satisfy 〈x, y〉 = 0, and x, y are both nonzero, then x and y

are linearly independent.(j) if Tn+1 = 0, then also Tn = 0.

Problem 2. Let V be a finite-dimensional space, U ⊂ V a subspace with U 6= V .• Suppose that x1, . . . , xr is a linearly independent list in V , so that

span(x1, . . . , xr) + U = V.

Prove that we can find xr+1, . . . , xn ∈ U so that x1, . . . , xn is a basisfor V .

• Prove there exists a basis (e1, . . . , en) for V so that ei /∈ U for all i.Problem 3. Suppose V,W are finite dimensional vector spaces. Let K be a subspace of

V and I a subspace of W .• Explain why {T ∈ L(V,W ) : null(T ) = K and image(T ) = I} is not a

subspace of L(V,W ).• Compute (with proof) the dimension of

{T ∈ L(V,W ) : null(T ) ⊃ K and image(T ) = I}

Problem 4. Let V be a finite-dimensional vector space over C. Let T ∈ L(V, V ).• Define the adjoint map T̂ : V ∗ → V ∗.• Suppose that there exists a basis of eigenvectors for T . Prove that

there exists a basis of eigenvectors for T̂ .• Suppose that image(T ) ⊂ null(T ). Prove that image(T̂ ) ⊂ null(T̂ ).

Problem 5. Let V be a finite dimensional vector space over the complex numbers. LetT ∈ L(V, V ). Let I be the identity transformation from V to V , and let

1

2

T, T 2, . . . be the successive powers of T . In this question we will considerthe subspace Q of L(V, V ) spanned by the (infinite) list 1, T, T 2, . . . .

• Compute Q in the case when V = C2 and T =(

2 00 3

).

• Prove that there exists N ≥ 1 so that {1, T, T 2, . . . , TN} spans Q.? Prove that dim(Q) ≤ dim(V ) always. 1

Problem 6. Consider R3 with the usual inner product (the dot product) and let M bea 3× 3 symmetric matrix, thought of as a linear map R3 → R3.

Let e1 = (1, 1, 2), e2 = (3, 1, 0) and let W be the span of e1, e2.• Find a vector e3 perpendicular to W .• Suppose you are given that M(w) ∈ W whenever w ∈ W , i.e., M

preserves W . Prove that e3 is an eigenvector of W .• Find the vector in W that is closest to (0, 0, 1).

Problem 7. Let A =(

3 21 2

).

• Compute the determinant, the characteristic polynomial, and eigen-values of A.

• Compute A100. (Please explain your procedure clearly; then you canget partial credit even if you make numerical errors.)

? Let B be an invertible 3× 3 complex matrix with the following prop-erty: for any integer n, positive or negative, all the matrix entries Bn

ij

of Bn satisfy |Bnij | ≤ 10. Prove that all the eigenvalues of B have

absolute value 1, and that B has a basis of eigenvectors.Problem 8. ? Suppose that V,W are finite-dimensional inner product spaces, and T :

V →W a linear map.• Define the singular values of T .? Suppose we are given orthonormal vectors v1, . . . , vk ∈ V – not neces-

sarily a basis! – so that ‖Tvi‖ ≥ ‖vi‖ for each i. Prove that at least kof the singular values of T – counted with multiplicity – are ≥ 1.

1Hint: Use the Cayley-Hamilton theorem.