Basis and Dimension: Understanding Vector Spaces and Bases in Linear Algebra, Assignments of Linear Algebra

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Basis and Dimension

MATH 322, Linear Algebra I

J. Robert Buchanan

Department of Mathematics

Spring 2007

Introduction

Today we will explore the concepts of

generalized coordinate systems, and

the dimension of a (finite dimensional) vector space.

Basis Vectors

Definition

If V is a vector space and S = { v 1 , v 2 ,... , v n } is a set of

vectors in V , then S is a basis for V provided,

(^1) S is linearly independent, and

(^2) V = span( S ).

Theorem

If S = { v 1 , v 2 ,... , v n } is a basis for V , then every v ∈ V can be

expressed as

v = c 1 v 1 + c 2 v 2 + · · · + cn v n

where the ci ’s are unique.

Proof.

Basis Vectors

Definition

If V is a vector space and S = { v 1 , v 2 ,... , v n } is a set of

vectors in V , then S is a basis for V provided,

(^1) S is linearly independent, and

(^2) V = span( S ).

Theorem

If S = { v 1 , v 2 ,... , v n } is a basis for V , then every v ∈ V can be

expressed as

v = c 1 v 1 + c 2 v 2 + · · · + cn v n

where the ci ’s are unique.

Proof.

Coordinates Relative to a Basis

Definition

If S is a basis for V and v ∈ V is written as

v = c 1 v 1 + c 2 v 2 + · · · + cn v n then the vector ( c 1 , c 2 ,... , cn ) ∈ R

n

is called the coordinate vector of v relative to S.

Notation: ( v ) S = ( c 1 , c 2 ,... , cn )

Remark: ( c 1 , c 2 ,... , cn ) is an ordered n -tuple and the order is

significant.

Coordinates Relative to a Basis

Definition

If S is a basis for V and v ∈ V is written as

v = c 1 v 1 + c 2 v 2 + · · · + cn v n then the vector ( c 1 , c 2 ,... , cn ) ∈ R

n

is called the coordinate vector of v relative to S.

Notation: ( v ) S = ( c 1 , c 2 ,... , cn )

Remark: ( c 1 , c 2 ,... , cn ) is an ordered n -tuple and the order is

significant.

Examples

Example

The standard basis for V = R

n is S = { e 1 , e 2 ,... , e n } where

e 1 =

, e 2 =

, · · · , e n =

Example

Determine if the set S = {( 1 , 1 , 1 ), ( 0 , 1 , 1 ), ( 1 , 0 , 1 )} is a basis

for V = R

3 .

Example

Find the coordinates of ( 3 , 4 , 5 ) relative to S.

Examples

Example

The standard basis for V = R

n is S = { e 1 , e 2 ,... , e n } where

e 1 =

, e 2 =

, · · · , e n =

Example

Determine if the set S = {( 1 , 1 , 1 ), ( 0 , 1 , 1 ), ( 1 , 0 , 1 )} is a basis

for V = R

3 .

Example

Find the coordinates of ( 3 , 4 , 5 ) relative to S.

Finite-dimensional Vector Spaces

Definition

A nonzero vector space V is finite-dimensional is a finite set

of vectors { v 1 , v 2 ,... , v n } forms a basis for V. The zero vector

space is also considered finite-dimensional. Any other vector

space is infinite-dimensional.

Example

Determine whether the following vector spaces are

finite-dimensional or infinite-dimensional. Be prepared to justify

your designation.

(^1) R n

(^2) P , the set of all polynomials with real coefficients

Finite-dimensional Vector Spaces

Definition

A nonzero vector space V is finite-dimensional is a finite set

of vectors { v 1 , v 2 ,... , v n } forms a basis for V. The zero vector

space is also considered finite-dimensional. Any other vector

space is infinite-dimensional.

Example

Determine whether the following vector spaces are

finite-dimensional or infinite-dimensional. Be prepared to justify

your designation.

(^1) R n

(^2) P , the set of all polynomials with real coefficients

Dimension

Theorem

If V is a finite-dimensional vector space and { v 1 , v 2 ,... , v n } is

any basis for V , then

(^1) any subset of V with more than n vectors is linearly

dependent,

(^2) any subset of V with less than n vectors cannot span V.

Proof.

Dimension (continued)

Corollary

All bases for the same finite-dimensional vector space have the

same number of vectors.

Remark: this is one of the most important results of linear

algebra because it makes the following definition meaningful.

Definition

The dimension of a finite-dimensional vector space V , denoted

dim( V ) is defined to be the number of vectors in a basis for V.

The zero vector space is defined to have dimension zero.

Dimension (continued)

Corollary

All bases for the same finite-dimensional vector space have the

same number of vectors.

Remark: this is one of the most important results of linear

algebra because it makes the following definition meaningful.

Definition

The dimension of a finite-dimensional vector space V , denoted

dim( V ) is defined to be the number of vectors in a basis for V.

The zero vector space is defined to have dimension zero.

Examples

Example

Determine the dimension of each of the following vector

spaces.

(^1) R n

2 M

mn

(^3) P n