3.4. THE DUAL OF A VECTOR SPACE AND MATRICES
181
Thus the
j
th
column of
OEST Ł
D;B
and of
OESŁ
D;C
OET Ł
C;B
agree.
n
For a vector space
V
over a field
K
, we denote the set of
K
–linear
maps from
K
to
K
by End
K
.V /
. Since the composition of linear maps is
linear, End
K
.V /
has a product
.S; T /
7!
ST
. The reader can check that
End
K
.V /
with the operations of addition and and composition of linear
operators is a ring with identity. To simplify notation, we write
OET Ł
B
in
stead of
OET Ł
B;B
for the matrix of a linear transformation
T
with respect to
a single basis
B
of
V
.
Corollary 3.4.12.
Let
V
be a finite dimensional vector space over
K
. Let
n
denote the dimension of
V
and let
B
be an ordered basis of
V
.
(a)
For all
S; T
2
End
K
.V /
,
OEST Ł
B;B
D
OESŁ
B
OET Ł
B
.
(b)
T
7!
OET Ł
B
is a ring isomorphism from
End
K
.V /
to
Mat
n
.K/
.
Lemma 3.4.13.
Let
B
D
.v
1
; : : : ; v
n
/
and
C
D
.w
1
; : : : ; w
n
/
be two
bases of a vector space
V
over a field
K
. Denote the dual bases of
V
by
B
D
v
1
; : : : ; v
n
and
C
D
w
1
; : : : ; w
n
. Let
id
denote the identity
linear transformation of
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Matrices, Vector Space, Linear map, ŒT B

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