Midterm
2
Review
Section 2.2.
V, W
are finite dimensional vector spaces with ordered bases
β, γ
, and
T, U
:
V
→
W
are linear transformations
(a)
For any scalar
a
,
aT
+
U
is a linear transformation from
V
to
W
.
•
True  Theorem 2
.
7(a), or just check that
(
aT
+
U
)(
cv
+
w
) =
c
(
aT
+
U
)(
v
) + (
aT
+
U
)(
w
)
(b)
[
T
]
γ
β
= [
U
]
γ
β
implies that
T
=
U
.
•
True  linear transformations are completely determined on a given basis
(c)
If
m
= dim(
V
) and
n
= dim(
W
), then [
T
]
γ
β
is an
m
×
n
matrix.
•
False  [
T
]
γ
β
is an
n
×
m
matrix (remember that [
v
]
β
is an
m
×
1 matrix
and [
T
(
v
)]
γ
is an
n
×
1 matrix)
(d)
[
T
+
U
]
γ
β
= [
T
]
γ
β
+ [
U
]
γ
β
.
•
True  Theorem 2
.
8(a)
(e)
L
(
V, W
) is a vector space.
•
True  see the two definitions on page 82
(f)
L
(
V, W
) =
L
(
W, V
).
•
False  if
V
and
W
are different sets, then a function from
V
to
W
cannot
be a function from
W
to
V
 see the definition of functions
Section 2.3.
V, W, Z
are finite dimensional vector spaces with ordered bases
α, β, γ
, and
T
:
V
→
W
and
U
:
W
→
Z
are linear transformations, and
A
and
B
are matri
ces
(a)
[
UT
]
γ
α
= [
T
]
β
α
[
U
]
γ
β
.
•
False  [
UT
]
γ
α
= [
U
]
γ
β
[
T
]
β
α
(see Theorem 2
.
11)
(b)
[
T
(
v
)]
β
= [
T
]
β
α
[
v
]
α
for all
v
∈
V
.
•
True  Theorem 2
.
14
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(c)
[
U
(
w
)]
β
= [
U
]
β
α
[
w
]
β
for all
w
∈
W
.
•
False  [
U
]
β
α
doesn’t even make sense since
α
is a basis for
V
, not
W
(d)
[
I
V
]
α
=
I
.
•
True  definitions
(e)
T
2
β
α
=
[
T
]
β
α
2
.
•
False 
T
2
doesn’t make sense unless
W
=
V
(f)
A
2
=
I
implies that
A
=
I
or
A
=

I
.
•
False 
(
1
0
0

1
)
2
=
I
(g)
T
=
L
A
for some matrix
A
.
•
False 
T
is defined on
V
,
L
A
is defined on
F
n
(h)
A
2
= 0 implies that
A
= 0, where 0 denotes the zero matrix.
•
False  (
0 1
0 0
)
2
= (
0 0
0 0
)
(i)
L
A
+
B
=
L
A
+
L
B
.
•
True  (
A
+
B
)
v
=
Av
+
Bv
(j)
If
A
is square and
A
ij
=
δ
ij
for all
i
and
j
, then
A
=
I
.
•
True  definition of the Kronecker delta (page 89)
Section 2.4.
V, W
are finite dimensional vector spaces with ordered bases
α, β
,
T
:
V
→
W
is a linear transformation, and
A
and
B
are matrices
(a)
[
T
]
β
α

1
=
T

1
β
α
.
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 Spring '08
 Serra
 Linear Algebra, Vector Space

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