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LECTURE 9
LINEAR VECTOR SPACES II
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LECTURE 9
LINEAR VECTOR SPACES II
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THE BASIS
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DEF.
S is a spanning set for space V if every vector in V is a linear
combination of vectors in S.
Example:
lin( (1,1,1), (2,2,2), (0,1,0), (0,3,0) ) = lin( (1,1,1) (0,1,0) )
The minimal spanning set must consist of independent vectors.
Spanning sets can contain redundant vectors. Every spanning list in a
vector space can be reduced.
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This reduced list is called the basis of the vector
space.
A consequence of the axiom of choice is that every vector
space has a vector basis.
BASIS
DEF.
A linearly independent spanning set for a vector space V is called
a basis for V.
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Problem
Does the set of vectors S = {(1,2), (3,1)} constitute a
basis for R
2
?
For an arbitrary vector w = (w
1
,w
2
) there exists a solution of the equation:
c
1
(1,2) + c
2
(3,1)= w,
with the equivalent form:
.
1
2
3
1
2
1
2
1
w
w
c
c
The determinant of the coefficient matrix is not 0, so the vectors are
linearly independent and neither is redundant.
The unit vectors e
1,
e
2
, e
3
, ..., e
n
in R
n
are a basis for R
n
,
they are called the standard basis for R
n
.
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Let
V
be a subspace of
R
n
, and let
B
= {b
1
, b
2
, ...,b
n
} be a subset of
V
.
Then the following statements are eqiuvalent:
1. B
is a basis for
V
2. B
is a minimal spanning set for
V
3. B
is a maximal linearly independent subset of
V
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DIMENSION OF A LINEAR SPACE
Definition
Let n vectors { v
1
, . . ., v
n
} constitute the basis of V. We then
say that the dimension of V is n and denote it as
dim V = n.
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Fact
Let A(a
1
, a
2
, ....., a
n
) denote an n
x
n matrix in which the i-
th column is vector a
i
.
If det A(a
1
, a
2
, ....., a
n
) 0, then the system { a
1
, a
2
, .....,
a
n
}
constitutes o basis of R
n
.
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COORDINATES OF A VECTOR RELATIVE TO
A GIVEN BASIS
Theorem
Let vectors a
1
, a
2
, ....., a
n
form the basis for a linear
space V.
Every vector b from this space is a linear combination
of
a
1
, a
2
, ....., a
n
i.e.:
b = c
1
a
1
+ .....+ c
n
a
n
,
the coefficients c
1
, c
2
, ....., c
n
are uniquely
determined and are called the coordinates of vector
b with respect to this basis (relative to this basis).
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Show that the coordinates of a vector b relative to
basis B = {a
1
, a
2
, ....., a
n
} are uniquely determined
Let b = c
1
a
1
+ .....+ c
n
a
n
, and
b = d
1
a
1
+ .....+ d
n
a
n
, then
0 = (c
1
- d
1
) a
1
+ .....+ (c
n
- d
n
) a
n
,
and from linear independence
c
1
- d
1
= 0, c
2
- d
2
= 0, ..., c
n
- d
n
= 0
QED
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TRANSITION MATRIX
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TRANSITION MARTIX
Let us consider a linear space V of dimension n
and two different bases of this space:
B
1
= {v
1
,...,v
n
}, B
2
= {u
1
,...,u
n
}.
We write the basis vectors from B
2
as a linear
combination of the basis vectors from B
1
:
u
1
=
p
11
v
1
+
p
21
v
2
+...+
p
n1
v
n
;
.....
.....
.....
u
n
=
p
1n
v
1
+
p
2n
v
2
+...+
p
nn
v
n
.
n
2
1
nn
2
n
1
n
n
2
22
21
n
1
12
11
n
2
1
u
u
u
p
p
p
p
p
p
p
p
p
v
v
v
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Definition
The transition matrix from basis B
1
to basis B
2
is
matrix P= [p
ij
] in which the elements of the
columns are the coordinates of vectors from the
new basis B
2
relative to the old basis B
1
:
nn
n
n
n
n
p
p
p
p
p
p
p
p
p
P
2
1
2
22
21
1
12
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Example – Space R
3
We consider the following two bases:
B
1
= { e
1
, e
2
, e
3
} unit basis; old
B
2
= { (1,0,0), (1,1,0), (1,1,1) } = { u
1
, u
2
, u
3
} new
The transition matrix from B
1
to B
2
is
.
P
1
0
0
1
1
0
1
1
1
u
1
= 1 e
1
u
2
= 1 e
1
+ 1 e
2
u
3
= 1 e
1
+ 1 e
2
+ 1 e
3
1
0
0
1
1
0
1
1
1
.
1
0
0
1
1
0
1
1
1
1
0
0
0
1
0
0
0
1
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e
1
= 1 u
1
e
2
= -1 u
1
+ 1 u
2
e
3
= -1 u
2
+ 1 u
3
1
0
0
1
1
0
0
1
1
P
1
u
1
= 1 e
1
u
2
= 1 e
1
+ 1 e
2
u
3
= 1 e
1
+ 1 e
2
+ 1 e
3
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Fact
If P is a transition matrix from basis B
1
to basis B
2
,
then P is invertible and the inverse matrix P
-1
is the
transition matrix from B
2
to B
1
.
Fact
In the case when B
1
is the unit basis of R
n
, the
transition matrix from the unit basis B
1
to an
arbitrary basis is simply composed of the vectors
from the new basis set up as columns.
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Fact
If the coordinates of vector w relative to B
1
are
w
1
,...,w
n
, then its coordinates w
1
’,...,w
n
’ relative to B
2
satisfy:
where P
is the transition matrix from B
1
to B
2
.
,
w
w
w
P
w
w
w
n
,
n
,
,
2
1
1
2
1
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Example cont.
The coordinates of vector w = (3,1,0) relative to B
2
are:
.
P
0
1
2
0
1
3
1
0
0
1
1
0
0
1
1
0
1
3
1
Thus: w = 2 u
1
+
u
2
.
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RANGE SPACE
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RANGE SPACE
The range of a matrix A
mxn
is defined to be the subspace
R(A) of R
m
:
R(A) = {Ax : x R
n
} R
m
The spanning set for R(A) is the the set of basic columns in A
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RANK PLUS NULLITY THEOREM
dim R(A) + dim N(A) = n
for all mxn matrices
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Summary
of ‘RANK’
Let A be a nm matrix.
1. rank(A) = the number of nonzero rows in any row echelon form
equivalent to A
2. rank(A) = the number of leading elements in any row echelon form
equivalent to A
3. rank(A) = the number of basic columns in A
4. rank(A) = the size of the largest nonsingular submatrix in A
5. rank(A) = the number of independent columns in A
6. rank(A) = the number of independent rows in A
7. rank(A) = dim R(A)
8. rank(A) = dim R(A
T
)
9. rank(A) = n - dim N(A)
10.rank(A) = m - dim N(A
T
)
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