In the vector space V of ( 2 × 2 ) matrices, let Q = { A 1 , A 2 , A 3 , A 4 } where A 1 = [ 1 0 0 0 ] , A 2 = [ 1 − 1 0 0 ] , A 3 = [ 0 2 0 0 ] , and A 4 = [ − 3 0 2 1 ] . Use the corollary to theorem 5 and the natural basis for V to show that Q is a basis for V .
In the vector space V of ( 2 × 2 ) matrices, let Q = { A 1 , A 2 , A 3 , A 4 } where A 1 = [ 1 0 0 0 ] , A 2 = [ 1 − 1 0 0 ] , A 3 = [ 0 2 0 0 ] , and A 4 = [ − 3 0 2 1 ] . Use the corollary to theorem 5 and the natural basis for V to show that Q is a basis for V .
Solution Summary: The author explains that the set Q is a basis for V.
In the vector space
V
of
(
2
×
2
)
matrices, let
Q
=
{
A
1
,
A
2
,
A
3
,
A
4
}
where
A
1
=
[
1
0
0
0
]
,
A
2
=
[
1
−
1
0
0
]
,
A
3
=
[
0
2
0
0
]
, and
A
4
=
[
−
3
0
2
1
]
.
Use the corollary to theorem
5
and the natural basis for
V
to show that
Q
is a basis for
V
.
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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