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
 
 alinear 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).