Answer:
Explanation:
1. Find a basis for the row space, column space, and null space of the matrix given
below:
A =
3 4 0 7
1 −5 2 −2
−1 4 0 3
1 −1 2 2
Solution. reef(A) =
1 0 0 1
0 1 0 1
0 0 1 1
. Thus a basis for the row space of A is
{[1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1]}. Since the first, second, and third columns of
rref(A) contain a pivot, a basis for the column space of A is {
3
1
−1
1
,
4
−5
4
−1
,
0
2
0
2
}.
If we solve Ax = 0, we find that x4 is a free variable, so we set x4 = r. We
obtain
x1
x2
x3
x4
= r
−1
−1
−1
1
, so {
−1
−1
−1
1
} is a basis for the nullspace of A.
2. What is the maximum number of linearly independent vectors that can be found
in the nullspace of
A =
1 2 0 3 1
2 4 −1 5 4
3 6 −1 8 5
4 8 −1 12 8
Solution. rref(A) has three columns with pivots and two columns without
pivots. Thus the dimension of the nullspace of A is 2, so at most 2 linearly
independent vectors can be found in the nullspace of A.
3. Let T : R3 → R3 be the linear transformation defined by
T([x1, x2, x3]) = [2x1 + 3x2, x3, 4x1 − 2x2].
Find the standard matrix representation of T. Is T invertible? If so, find a
formula for T
−1
.
Mathematics Department 1