Math 235 Practice Problems 1 1. Find a basis for the four fundamental subspace of each matrix. −2 2 1 1 2 −3 −4 1 −2 1 3 −5 3 1 2 −1 −2 3 4 0 −4 2 . (a) A = (b) A = . (c) A = −1 2 1 7 4. 0 1 −1 −3 1 −2 −2 4 4 0 4 1 2 3 −5 −5 0 1 2. Let A = and define L : M2×2 (R) → M2×2 (R) by L (X) = AX − XA. 1 0 (a) Prove that L is linear. 1 2 (b) Evaluate L . −3 1 3 −1 (c) Find a matrix C ∈ M2×2 (R) such that L(C) = . 1 −3 3. Invent L : R3 → P 2 (R) satisfies that alinear mapping 0 0 1 L 0 = x2 , L 1 = 2x, L 0 = 1 + x + x2 . 1 0 0 4. Prove that if L : V → W is a linear mapping, then the mapping tL : V → W defined by (tL)(~x) = tL(~x) is linear. 5. Let A ∈ Mm×n (R) and B ∈ Mn×m (R). (a) Prove that rank(BA) ≤ rank(B). (b) Prove that dim Null(B) ≤ dim Null(AB). (c) Prove that rank(AB) ≤ rank(B). 6. Let A ∈ Mm×n (R). Prove that every vector in Row(A) is orthogonal to every vector in Null(A). 2 7. Determine whether each a counter example. 1 (a) If the RREF of A is 0 0 statement is true or false. Justify your answer with a proof or 0 0 0 0 1 0 , 1 . 1 0 0 , then a basis for Col(A) is B = 0 0 0 0 0 1 0 1 0 0 0 0 1 T (b) If the RREF of A is 0 1 0 0 , then a basis for Col(A) is , . 0 0 0 0 0 0 0 0 (c) There exists a 3 × 3 matrix A such that rank(A) = dim Null(A). (d) Let A be an m × n matrix and let {~v1 , . . . , ~vk } be a basis for Null(A). If we extend {~v1 , . . . , ~vk } to a basis {~v1 , . . . , ~vk , ~vk+1 , . . . , ~vn } for Rn , then {~vk+1 , . . . , ~vn } is a basis for Row(A).