L 9 Linear Spaces II

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1

LECTURE 9

LINEAR VECTOR SPACES II

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2

THE BASIS

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3

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

11

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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 nm 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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