MATH 110: LINEAR ALGEBRA HOMEWORK #4 WORKSHEET

JOHN VOIGHT

Work on exercise assigned to your group (by number) at the board, and then write up together a nice solution that everyone can understand. If you finish early, return to your desks and work together on another problem of your choice.

(1) [2.2.10] Let V be a vector space (over a field F). Let W_{1}, W_{2} be subspaces such that V = W_{1} ⊕ W_{2}. Let T : V → V be a linear transformation such that T (W_{1}) ⊂ W_{1} and T (W_{2}) ⊂ W_{2}. Prove that there is an ordered basis β of V such that [T]_{β} has the form

(\begin{matrix}A_{1} & O \\ O & A_{2} \end{matrix}) ,

where A_{1} and A_{2} are matrices. What are the dimensions of the matrices A_{1} and A_{2}? Prove that T (W_{1}) = W_{1} if and only if A_{1} is an invertible matrix.

(2) [2.2.14] Let V = P (ℝ), and for j ≥ 1 define T_{j} : V → V by T_{j}(f(x)) = f^{(j)}(x), where f^{(j)}(x) is the jth derivative of f(x). Prove that the set {T_{1}, T_{2}, …, T_{n}} is a linearly independent subset of L(V) for any positive integer n.

(3) [2.3.11] Let V be a vector space, and let T : V → V be linear. Prove that T^{2} = T_{0} (the zero transformation) if and only if R(T) ⊂ N(T). Prove that if one of these two equivalent conditions holds, and V is finite-dimensional, then there is an ordered basis β for V such that [T]_{β} has the form

(\begin{matrix}O & A \\ O & O \end{matrix})

for some matrix A.

(4) [2.3.18(a)] Let V be a finite-dimensional vector space, and let T : V → V be linear. If rk(T) = rk(T^{2}), then prove that R(T)∩N(T) = {0}. Deduce that V = R(T)⊕N(T).

(5) [2.4.15] Let V and W be finite-dimensional vector spaces, and let T : V → W be a linear transformation. Suppose that β is a basis for V. Prove that T is an isomorphism if and only if T (β) is a basis for W.

(6) “Label” true or false:

(a) Any two vector spaces of the same (finite) dimension are isomorphic.

(b) The set of n × n invertible matrices is a subspace of M_{n×n}(F).

(c) If A is a matrix, then A^{2} = A implies A = I or A = O (the identity and zero matrix, respectively).

(d) If T : V → W is an invertible linear transformation, and β is a basis for V, γ a

basis for W, then ([T]_{β}^{γ})^{−1} = [T^{−1}]_{β}^{γ}.

(7) Let B be an n × n matrix. Prove that the map T_{B} : M_{n×n}(F) → M_{n×n}(F) by A ↦ AB is an isomorphism if and only if B is invertible.

Date: February 9, 2004.