Wednesday February 4
−
Lecture 14:
Abstract vectors spaces and subspaces
(Refers
to section 4.1)
Expectations:
1.
Define
abstract vector space
.
2.
Define closure under an operation.
3.
Prove:
a
) 0 ·
v
=
0
, for all
v
in
V
,
b
)
α
·
0
=
0
for all
α
in
R
.
c
) (
−
1)·
v
=
−
v
, for
all
v
in
V.
4.
Prove: If
α
in
R
and
v
in
V
are such that
α
v
=
0
. Then either
α
= 0 or
v
=
0
.
5.
Recognize
R
n
, the set of all
m
by
n
matrices,
M
n,m
,
and all functions
f
on [0,1] as
examples of vector spaces.
14.1
Definitions
−
An
abstract
vector space
, (
V
, +, scalar multiplication), is a set
V
along
with two operations + and scalar multiplication that satisfy all the axioms in the list
Vector space Axioms
.
14.1.1
Remark – In the above definition we use the real numbers as scalars. This is
why we often specify “vector space
over the reals
”.
•
The means that we only consider real numbers as scalars. We sometimes
speak of a “vector space
over the complex numbers
” as a way of saying that
we accept complex numbers as scalars.
14.1.2

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