Homework 10 Solution

In each of the following you are given two vector spaces and a function between them. Determine whether the function is a linear transformation or not. Prove your claim.



i.

T : R3 7!M2(R)




given by,



0
x
1


x + y y
2z
T
y
=
3x + z
0


@
z
A
(


)










ii.




: R2[x] 7!R3
given by,

0 1

p(2)

T p = @ p′ (2) A




p′′ (2)




iii.

T : M2(R) 7!M2(R)




given by,

TA=A2




iv.

: R2[x] 7!R



given by,
∫0 1 p(x)dx
T p =
Fix B 2 M3(R) and consider the function:



T : M3(R) 7!M3(R)




given by,




TA=AB




vi.

: M2(R) 7!M2(R)



given by,

T (
a b
) = (
a
c
+
1
2a + 3b + 2
)
c d
d
b
8
2a



1






2




In each of the following you are given a linear transformation (you don't need to prove that it is a linear transformation). Follow the following directions for each such transformation:



Find a basis for the kernel and the image of this transformation.



Find the dimension of the kernel and the image of this transformation. (Remark: This question will continue in the next HW, you may want to keep a copy of your solution to this part of the question).
Determine whether the transformation is onto. Explain your answer.



d. Determine whether the transformation is 1 1. Explain your answer. i. For






B
1
2
1
C
A =
0
5
5
0
3
1
2
1


B






C


@
1
7
6
A









consider the operator




TA : R3 7!R4

where the notation TA was de ned in class.

ii.

: M2(R) 7!R2



given by
(


)
SA=A
32
iii.

L : R3[x] 7!R2




given by

Lp = (
p(2) p(1)
)
p′ (0)
iv.




: R3 7!R3[x]

given by

0 1

a




@ b A = (a + b) + (a 2b + c)x + (b 3c)x2 + (a + b + c + d)x3 c

Let V; W be vector spaces over R and T : V 7!W be a linear operator. Let v1; :::; vn 2 V . For each of the following claims determine whether it is true or false. Prove or disprove your claim accordingly.



If v1; :::; vn is linearly independent in V then T v1; :::; T vn is linearly independent in W .



If T v1; :::; T vn is linearly independent in W then v1; :::; vn is linearly independent in V .






3




iii. If v1; :::; vn is a spanning set in V then T v1; :::; T vn is a spanning set in

.
iv. If T v1; :::; T vn is a spanning set in W then v1; :::; vn is a spanning set in

.



v. If U is a subspace of V then the set fT u : u 2 Ug is a subspace of W .




v. If U is a subset of V and the set fT u : u 2 Ug is a subspace of W then

U is a subspace of V .




Consider the claims which you determined are false in Q5. Which of them will become true if you add one of the following conditions? Prove your answers.



If you add the condition that T is 1-1 (but not necessarily onto).



If you add the condition that T is onto (but not necessarily 1-1).



Prove or disprove the following claims:



(Hint: You may want to use your answers to Q4 while solving this ques-tion. Alternatively, material which we will study in class next week may help.)

There exists T : M2(R) 7!R3 which is 1-1.



There exists T : R3 7!M2(R) which is 1-1.



There exists T : M2(R) 7!M2(R) which is neither 1-1 nor onto.



There exists T : M2(R) 7!R3[x] which is 1-1 but not onto.



There exists T : M2(R) 7!R3 which is onto but not 1-1.



There exists T : R3 7!M2(R) which is onto but not 1-1.



There exists T : M2(R) 7!R3 which neither onto nor 1-1.