Review Problem Set 2 (Vector Spaces)
Math 2270-001 (Summer 2005)
Do NOT Sumbit
Name:
1
ID:
Code Name:
Instructions
1. Print your name, ID number, and code name above. Your code name will be used for posting grades on the course
web page. If you do not supply a code name, your grades will not be posted.
2. If you supplied a code name in the last Test, please leave the code name entry blank.
3. Time allowed: 1 hour.
4. Solve all problems and show all work for full credit.
5. No calculators are allowed.
6. The total number of points in this test is 100. It will be graded out of 100.
1. Suppose E = (~e1 ; : : : ;~e3 ) and F = (~f1 ; : : : ; ~f2 ) are bases for R3 and R2 respectively. Let T 2 L(R3 ; R2 ) be given
by:
T (1~e1 + 2~e2 + 3~e3 ) := (11 1 + 12 2 + 13 3 ) ~f1 + (21 1 + 22 2 + 23 3 ) ~f2 :
Find MF (T (E )).
2. a) Give an example of an innite-dimensional vector space and prove that it indeed has innite dimension.
b) Give an example of a non-linear map between two vector spaces.
3. a) What is the dimension of Rmn as a vector space over R?
b) What is the relation between dim L(Rn ; Rm ) and dim(Rmn )?
c) Let f : Rn ! R be a non-zero linear map. Find dim(ker(f )).
4. Let T 2 L(U; V ) and S 2 L(V; W ), where U , V , W are nite-dimensional vector spaces, and let E , F , G be bases,
respectively, for U , V , W . Suppose
2 3 1 3 13
" 2 3 2#
4
5
2 1 2 0 ; and MG (S (F )) =
MF (T (E )) =
:
2 2 1
2 1 1 0
Find MG (S T (E )).
5. Prove or disprove the linear independence of the following vectors:
2 13 2 43 2 63
66 1 77 ; 66 2 77 ; 66 0 77 :
4 45 4 45 4 45
8
2
3
6. Prove that the image of a linear map is always a vector subspace of the codomain vector space.
7. Find a basis for the kernel of the linear map f : R4 ! R3 dened by:
0x
1
B
x
B
f @ x2
3
x4
1 2
1
C
C
4
:=
0
A
2
2
5
1
1
2
4
0x
3
4 B x1
2 5B
@ x2
10
3
x4
1
CC :
A
Review Problem Set 2 (Vector Spaces)
Math 2270-001 (Summer 2005)
Do NOT Sumbit
8. Let T : R3 ! R2 be dened by
02 x 31 T @4 y 5A := 3xx +52yy+ 34zz
Let E and B be two bases for R3 given by:
z
82 1 3 2 0 3 2 0 39
<
=
E := :4 0 5 ; 4 1 5 ; 4 0 5; ;
0
0
1
and
82 1 3 2 1 3 2 1 39
<
=
B := :4 1 5 ; 4 1 5 ; 4 0 5; :
1
Let F and C be two basis for R2 given by:
F :=
a)
b)
c)
Find ME (B) and MF (C ).
Find MF (T (E )).
Find MC (T (B)).
2
1 0 0 ; 1
; and
C :=
0
1 2 3 ; 5
0
:
Review Problem Set 2 (Vector Spaces)
Math 2270-001 (Summer 2005)
Do NOT Sumbit
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