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ARE211, Fall 2007
LINALGEBRA2: TUE, OCT 9, 2007
PRINTED: OCTOBER 18, 2007
(LEC# 12)
Contents
3.
Linear Algebra (cont)
1
3.6.
Vector Spaces
1
3.7.
Spanning, Dimension, Basis
4
3.8.
Matrices and Rank
8
3.
Linear Algebra (cont)
3.6.
Vector Spaces
A key concept in linear algebra is called a
vector space.
What is it? It is a special kind of set of
vectors, satisfying a particular property. The property is a bit abstract, so we’ll work up to it:
Verbal Defn:
a nonempty set of vectors
V
is called a
vector space
if it satisFes the following property:
given any two vectors that belong to the set
V
,
every
linear combination of these vectors is also in
the space.
One kind of vector space is a set of vectors represented by a line. Note that a line can (and should)
be thought of as just another set of vectors. Think about all of the lines you can draw in
R
2
. Which
lines are vector spaces, according to the above deFnition?
1
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LINALGEBRA2: TUE, OCT 9, 2007
PRINTED: OCTOBER 18, 2007
(LEC# 12)
•
verbal defn excludes any straight line that doesn’t go thru the origin.
•
verbal defn excludes any line that “curve.”
•
verbal defn excludes any straight line thru the origin that “stops.”
What’s left is rays thru the origin. Every ray through the origin is a vector space. Mathematically,
any ray through the origin is deFned as the set of all scalar multiples of a given vector. It’s called a
“onedimensional vector space”, because it is “constructed” from a single vector. Note that neither
the nonnegative or the positive cones deFned by a single vector are vector spaces.
Turns out that there’s exactly two more vector spaces in
R
2
:
•
The set consisting of the entire plane is a vector space, i.e.,
R
2
. This is a two dimensional
vector space.
•
The set consisting of zero alone is a vector space. This is a zero dimensional vector space.
Math Defn:
a nonempty set of vectors
V
is called a
vector space
if for all
x
1
,x
2
∈
V
, for all
α
1
,α
2
∈
R
,
α
1
x
1
+
α
2
x
2
∈
V
.
See how this deFnition stacks up against our examples:
•
look at a line that curves: take any vector in the
set
consisting of the points in the line.
Let
v
1
,
v
2
be any vectors in the line. Take zero times the Frst and 2 times the second. Is
it in the set? No. Conclude that the curved line is not a vector space.
•
look at a straight line that doesn’t pass through the origin. Let
v
1
,
v
2
be any vectors in
the line. Take zero times both. In the set? No. Conclude that a straight line that doesn’t
pass through the origin is not a vector space.
ARE211, Fall 2007
3
•
take any straight line thru the origin that “stops.” Let
v
1
,
v
2
be any vectors in the line.
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.
 Fall '07
 Simon

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