Which of the following transformations are linear transformations? Justify
your answer.
(i) ‘ : Mm⇥m(R) ! Mm⇥m(R) given by ‘(A) = A
“1
.
(ii) ‘ : P2
! P3 given by
‘(f(x)) = x
3
f
0
(0) +x
2
f(0),
where f
0
(x) is the first derivative of f(x).
Problem 2 (2 points)
Let ‘ : R
2
! R
2
be a linear transformation such that
‘
✓
1
1
◆
=
✓
1
“2
◆
and ‘
✓
“1
1
◆
=
✓
2
3
◆
.
(i) Find ‘
✓
“1
5
◆
.
(ii) Find ‘
✓
x
y
◆
.
Problem 3 (2 points)
Let ‘ : P2
! P3

be a linear transformation for which we know
‘(1) = 1, ‘(x) = x
2
, and ‘(x
2
) = x+x
2
.
(i) Find ‘(3″5x+ 2x
2
).
(ii) Find ‘(c+bx+ax2
).
1Problem 4 (2 points)
Let ‘ : M2⇥2(R) ! M2⇥2(R) be the linear transformation given by
‘(A) = ✓
1 2
3 4 ◆
A
for every A 2 M2⇥2(R). Consider the bases
S =
⇢✓ 1 0
0 0 ◆
,
✓
0 1
0 0 ◆
,
✓
0 0
1 0 ◆
,
✓
1 0
0 1 ◆$
and
T =
⇢✓ 1 0
0 1 ◆
,
✓
1 1
0 0 ◆
,
✓
1 0
1 0 ◆
,
✓
0 1
0 0 ◆$
for M2⇥2(R). Find [‘]
S
T
and [‘]
T
S
.
Problem 5 (2 points)
Let ‘ : P1
! P2
be the linear transformation given by
‘(f(x)) = xf(x) +f(0).
Consider the following bases for P1 and P2:
S = {1 +x,”1 +x}
and
T = {1 +x,”1 +x,1 +x
2
}.
Find the matrix representation [‘]
T
S
with respect to S and T.
2