May 2024
Intermediate to advanced
687 pages
18h 35m
English
Content preview from Discrete Mathematical Structures by Pearson
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Vector Spaces 12-33
where
EE
12
and ′=
−
′=
−
10
10
01
01
, which shows that the matrix B ∈ S ∩ T is a linear
combination of matrices E
1
′ and E
2
′.
Further, E
1
′ and E
2
′ are linearly independent as l
1
E
1
′ + l
2
E
2
′ = 0
⇒
−
+
−
=
ll
12
10
10
01
01
00
00
⇒
−
+
=
l
l
l
l
1
1
2
2
0
0
0
0
00
00
⇒
−−
=
ll
ll
12
12
00
00
⇒ l
1
= l
2
= 0
Hence, dim (S ∩ T ) = 2, a basis being matrices
10
10
01
01
−
−
,
Therefore, dim (S + T ) = dim S + dim T - dim S ∩ T
= 3 + 3 - 2
= 4
12.6.1 Coordinates
A useful feature of a basis in a vector space V is to be able to introduce coordinates similar to
natural coordinates.
If
xxxx xe
ni
i
i
n
=…
()
=
∼
=
∑
12
,,,
1
Then {e
i
} is a standard basis. A given vector
x
has a unique representation as a linear
combination of the ...
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