Homework 11 Solution

Note: Several questions below are not mandatory for submission.




These are indicated explicitly.




1. Consider the following linear transformations:




S : M2(R) 7!R2







: R3[x] 7!R2






: R3 7!R3[x] given by




)
given by
3
SA=A

2




)
given by
Lp =
p(2) p(1)


p′(0)




1
a

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

c



TA:R2
7!R2
where
A = (
2
1
)


1
2
Determine which of the following compositions is de ned. If it is de ned then specify its domain, specify its codomain and nd a formula for the composition:

◦L;L◦ ;TA2;S ◦TA;TA ◦S;

◦TA;TA2 ◦S;TA ◦L◦ :




[There is no mandatory submission of this question. ] Complete the following parts of proofs of theorems from class:



If V; W are vector spaces and T : V ! W is a linear transformation then kerT is closed to multiplication by scalar.



If V; W are vector spaces and T : V ! W is a linear transformation then ImT is closed to multiplication by scalar.



Corollary of the dimension formula: If V; W are vector spaces such that



dimV =dimU and T : V ! W is a 1 1 linear transformation then T is onto.




Corollary of the dimension formula: If V; W are vector spaces such that



dimV =dimU and T : V ! W is an onto linear transformation then T is 1 1.




If V; W; U are vector spaces and T : V ! W , S : W ! U is a linear transformation then S ◦ T 'respects' multiplication by scalar.



In each of the following you are given a transformation between a space Rn and a space Rm for some n and m. Find a matrix A for this transformation such that the transformation is equal to multiplication by this matrix.



1






2




i. Let S : R2 7!R2 be the composition S3 ◦ S2 ◦ S1 where S1 is rotation by =4 radians counterclockwise, S2(x; y) = (x; y), and S3 is rotation by =4 radians clockwise.




The transformation TA ◦ L ◦ where TA; L; were de ned in Q1.



[In this question it is mandatory to submit only odd numbered questions (i,iii,v, and so on).]



i. There exists T : R2[x] 7!M2(R) such that 1 x + 2x2 is in its kernel
and (
0
1
) is in its image.


1
1





There exists T : R3 7!R5 such that
0 1

kerT = { @
x
A : x + y + z + w = 1}
y
z
There exists T : R3 7!R5 such that
0 1

kerT = { @
x
A : x + y + z = 0
}
y
z
For U = fp 2 R3[x] : p(1) = p′(1)g there exists an isomorphism T : U 7!M2(R).



For U = fp 2 R3[x] : p(1) = p′(1)g there exists an isomorphism T : U 7!R2[x].



There exists T : R4 7!R6 such that dimkerT =2dimImT .



There exists T : R4 7!R6 such that dimkerT =2dimImT +1.



Let V be a vector space over R such that dimV =3 and let T : V 7!V be a
linear transformation which satis es T 3 = 0 and T 2 ≠ 0. Prove:




i. There exists v 2 V which is not the zero vector such that T v = 0.




ImT kerT 2 and ImT 2 kerT .



kerT kerT 2 and kerT ≠kerT 2



dimkerT = 1.



6. Let V; W be two vector spaces over R. Assume that T : V 7!W and




S : W 7!V are linear transformations which satisfy:




S ◦ T = idV

Prove or disprove:

i. T is 1 1.




ii. S is 1 1.







3




dimW =dimV .



If dimW =dimV then S is an isomorphism.



If T is not onto then dimW ≠dimV .



[In this question it is mandatory to submit only odd numbered questions (i,iii,v, and so on).]



Consider the following linear transformations T; S : R3[x] 7!R3[x] given by
T p(x) = p′(x) and Sp(x) = p(x + 1)




and consider the following bases of R3[x]:




E = f1; x; x2; x3g




and




= f1; 1 + x; (1 + x)2; (1 + x)3g



Find [T ]BE, [T ]BB.



[T]EB, [T]EE.
Find [S]EE, [S]BB.



Find [T ◦ S]EE.



[T ◦ S]BB.
For each one of the following transformations choose a basis for the domain and the codomain of the transformation. Write the matrix that represents the transformation with respect to these bases, and use it to nd a basis for the kernel and image of the transformation.



The transformations L from Q1.



The transformations from Q1.



Consider the vector space,









W ={(
a b
) : a + d = 0}:






c d
Let H = (
0
1
) and consider the linear transformation L : W 7!W
1
0
de ned by LA = AH HA.




i. Prove that L indeed acts from W to W .




Find an ordered basis for W and denote it B.
Find [L]BB.



Use [L]BB to nd a basis for the kernel and image of L.






4

Consider the linear transformation S : R2[x] 7!R3 which is de ned by



0 1

p(0)

Sp = @ p(1) A

p(2)




i. Is S an isomorphism? Justify your answer.

ii. If your answer to part (i) was "yes" then nd S 1.




10. [In this question it is mandatory to submit only odd numbered questions (i,iii,v, and so on).]

i. Consider the following ordered bases of R3:

0 10 10 1



1


0


1


B=(@0
A;@ 1 A;@
1 A)


1


1


0


C=(0 2
1;0 1 1;0
0 1)
@
1
A @
2
A @
1
A
1
0
3
E=(00
1;0 1 1;0
0 1)
@
1
A @
0
A @
0
A
0
0
1
Find the following matrices of transition from basis to basis:




[id]BE; [id]CE; [id]EB; [id]EC; [id]BC; [id]CB:




Consider the transformations S and T and the bases B and E from Q7(a). Find the following matrices of transition from basis to basis:



[id]BE; [id]EB:




Check that the formula of transition from basis to basis holds in the following cases:




[T ]B = [id]EB[T ]E[id]BE




[S]E = [id]BE[S]B[id]EB




iii. Consider the vector space W and the linear transformation L in 8. Here

are two bases of W :
1 )
;
( 0
0 );(
1
0 )
)
B=((
0


1
0


0
1


0
0


1
0
) ; (


1
1
) ; (
1
1
)):
C=((0
1


1
1
1
1
Find [id]CB, [id]BC and check that the formula of transition from basis to basis holds in the following case:




[L]C = [id]BC[L]B[id]CB







5




11. Let V be a vector space such that dim V = 3. Assume that B = (v1; v2; v3) and C = (w1; w2; w3) be bases of V such that

1
2 0 2

B
1
0
0


[id]C = @
A
3
5
2
a. Are fw1; v2; v3g are linearly independent? Justify your answer. b. Is it true that w2 = v1 + v3 v2? c. Find [w1]B.