linear algebra
Linear Transformations and Rank-Nullity
August 17, 2020
Question 1
For the following three Linear Transformations: 1) find a basis for the range;
2) find the kernal; and 3) verify the rank-nullity theorem in each case.
a) T : R3[x] M22(R) given by
T (a + bx + cx2 + dx3) =
(
a b c
d a + c
)
b) T : M32(R) R3 given by
T (
a11 a12a21 a22
a31 a32
) =
a11 + a12a21 + a22
a31 + a32
c) T : R4 R3[x] by
T (
a
b
c
d
) = (a c) + (b c)x + (a b)x3
Question 2
Write the matrix associated to the 3 linear transformations above with re-
spect to the following given bases:
1
a)
BR3[x] = (1, x, x
2, x3)
CM22(R) =
((
1 0
0 0
)
,
(
0 1
0 0
)
,
(
0 0
1 0
)
,
(
0 0
0 1
))
b)
BM32(R) =
(1 00 0
0 0
,
0 10 0
0 0
,
0 01 0
0 0
,
0 00 1
0 0
,
0 00 0
1 0
,
0 00 0
0 1
)
CR3 =
(10
0
,
01
0
,
00
1
)
c)
BR4 =
(
1
0
0
0
,
0
1
0
0
,
0
0
1
0
,
0
0
0
1
)
CR3[x] =
(
1, x, x2, x3
)
Question 3
For the following, give an example if one exists, or state it is not possi-
ble. If it is not possible, explain why.
a) An injective linear transformation T : M33(R) R6[x]
b) A surjective linear transformation T : R3[x] M22(R)
c) An injective linear transformation between two vectors spaces of the same
dimension that is not an isomorphism.
d) A surjective linear transformation T : R5[x] R3 with nullity(T)= 4
e) An injective linear transformation T : R3 R4 with rank(T)=3
2
Question 4
State whether the following are true or false: if they are false give a counter
example.
a) Every linear transformation T : V W with dim(V ) < dim(W ) is injec-
tive.
b) Every surjective linear transformation T : V W with dim(V ) = dim(W )
is an isomorphism.
c) Every linear transformation T : V W with dim(V ) > dim(W ) is sur-
jective.
d) Suppose V = W (V is isomorphic to W). Then every linear transformation
T : V W is an isomorphism.
3
