linear algebra
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Math_21_HW_7__Linear_Transformations_and_Rank_Nullity2.pdf
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] −→ M2×2(R) given by
T (a + bx + cx2 + dx3) =
( a b− c d a + c
)
b) T : M3×2(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)
CM2×2(R) =
(( 1 0 0 0
) ,
( 0 1 0 0
) ,
( 0 0 1 0
) ,
( 0 0 0 1
))
b)
BM3×2(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 : M3×3(R) −→ R6[x] b) A surjective linear transformation T : R3[x] −→ M2×2(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.
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