May 2024
Intermediate to advanced
687 pages
18h 35m
English
Content preview from Discrete Mathematical Structures by Pearson
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Vector Spaces 12-7
Note 12.3 Let V be the set of real-valued functions over the domain [0, 1], then V is a vector
space over R with respect to addition and scalar multiplication of functions.
12.2.1 Some Simple Properties of a Vector Space
THEOREM 12.1 Let V( F ) be a vector space and
0
be the zero vector of V. We then have
(1)
aaF
00=∈,
(2)
00a
~
a
~
=∈
,
V
(3) aa aF V−
()
=−
()
∈∈
aa
~
a
~~
,,
(4)
aaaa
FV
ab ab ab−=
∈∈
(
)
− ,,,
(5)
aa a
or=⇒==
000a
Proof:
(1)
00 00+= ∈, V
Therefore,
aa
00+
(
)
= 0
aa aa
00 00 00+=+⇒=
By left cancellation law as V is an abelian group w. r. t. addition.
(2) 0 + 0 = 0, 0 ∈ F
Therefore,
00 0 +
(
)
=
aa
00 0000
aa aa+=+ =⇒
By left cancellation law as V is an abelian graph w. r. t. addition.
(3) Now
aa
a
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