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3.4. THE DUAL OF A VECTOR SPACE AND MATRICES
179
Corollary 3.4.9.
.
dim
W
C
dim
W
ı
D
dim
V
.
Proof.
Exercise
3.4.9
.
n
Matrices
Let
V
and
W
be ﬁnite dimensional vector spaces over a ﬁeld
K
. Let
B
D
.v
1
;:::;v
m
/
be an ordered basis of
V
and
C
D
.w
1
;:::;w
n
/
an
ordered basis of
W
. Let
C
±
D
.w
±
1
;:::;w
±
n
/
denote the basis of
W
±
dual
to
C
. Let
T
2
Hom
K
.V;W /
.
The
matrix
ŒT Ł
C;B
of
T
with respect to the ordered bases
B
and
C
is
the
n
–by–
m
matrix whose
.i;j/
entry is
h
T v
j
;w
±
i
i
.
Equivalently, the
j
–th column of the matrix
ŒT Ł
C;B
is
S
C
.T.v
j
//
D
2
6
6
6
4
h
T.v
j
/;w
±
1
i
h
T.v
j
/;w
±
2
i
:
:
:
h
T.v
j
/;w
±
n
i
3
7
7
7
5
;
the coordinate vector of
T.v
j
/
with respect to the ordered basis
C
.
Another useful description of
ŒT Ł
C;B
is the following:
ŒT Ł
C;B
is the
standard matrix
of
S
C
TS
²
1
B
W
K
m
±!
K
n
:
Here we are indicating com
position of linear maps by juxtaposition; i.e.,
S
C
TS
²
1
B
D
S
C
ı
T
ı
S
²
1
B
. As
discussed in Appendix
E
, the standard matrix
M
of a linear transformation
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Matrices, Vector Space

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