Let V be the vector space of ( 2 × 2 ) matrices and define T : V → V by T ( A ) = A T (see Exercise 9 of Section 5.8 ). Let B = { E 11 , E 12 , E 21 , E 22 } be the natural basis for V . a) Find the matrix, Q , of T with respect to B . b) For arbitrary A in V , show that Q [ A ] B = [ A T ] B . 9. Let V be the vector space of ( 2 × 2 ) matrices and define T : V → V by T ( A ) = A T . Show that T is invertible and give the formula for T − 1 .
Let V be the vector space of ( 2 × 2 ) matrices and define T : V → V by T ( A ) = A T (see Exercise 9 of Section 5.8 ). Let B = { E 11 , E 12 , E 21 , E 22 } be the natural basis for V . a) Find the matrix, Q , of T with respect to B . b) For arbitrary A in V , show that Q [ A ] B = [ A T ] B . 9. Let V be the vector space of ( 2 × 2 ) matrices and define T : V → V by T ( A ) = A T . Show that T is invertible and give the formula for T − 1 .
Solution Summary: The author explains the matrix Q for T with respect to B.
Let
V
be the vector space of
(
2
×
2
)
matrices and define
T
:
V
→
V
by
T
(
A
)
=
A
T
(see Exercise
9
of Section
5.8
). Let
B
=
{
E
11
,
E
12
,
E
21
,
E
22
}
be the natural basis for
V
.
a) Find the matrix,
Q
, of
T
with respect to
B
.
b) For arbitrary
A
in
V
, show that
Q
[
A
]
B
=
[
A
T
]
B
.
9. Let
V
be the vector space of
(
2
×
2
)
matrices and define
T
:
V
→
V
by
T
(
A
)
=
A
T
. Show that
T
is invertible and give the formula for
T
−
1
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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