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EE221A Section 7
10/7/11
1
Practice midterm
Problem 1.
Injectivity and surjectivity
a) Suppose that
T
:
V
→
W
is an injective, linear map, and that
{
v
1
,...,v
n
}
is a linearly independent set in
V
. Prove that
{
T
(
v
1
)
,T
(
v
2
)
,...,T
(
v
n
)
}
is a linearly
independent set in
W
.
b) True or false: If
L
:
X
→
Y
is a surjective, linear map, and
{
x
1
,...,x
n
}
spans
X
, then
{
L
(
x
1
)
,...,L
(
x
n
)
}
spans
Y
. Please support your answer by a proof or coun
terexample.
Problem 2.
Subspaces. True or false: If
X
1
and
X
2
are subspaces of a vector space
X
, then
X
1
∩
X
2
is also a subspace of
V
. Please support your answer by a proof or
counterexample.
Problem 3.
Matrix Representation of Linear Maps (4 points). Consider the vector
space
R
2
, and the linear map
R
θ
which takes any vector in
R
2
and rotates it in a
counterclockwise direction about the origin through an angle
θ
.
(a) Derive the matrix representation of this linear map with respect to the standard
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.
 Fall '10
 ClaireTomlin

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