Vector Spaces

Vector Spaces

Shahid Hussain

March 16, 2009

Shahid Hussain () Vector Spaces March 16, 2009 1 / 17

Outline

1 Spaces and SubspacesSpacesSubspaces


Spaces and Subspaces Spaces

Introduction

Matrix theory became established toward the end of the nineteenthcentury,

Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:

Points in the plane R2,points in 3-space R3,polynomials,continuous functions, anddifferentiable functions (to name only a few)

were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.

Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.

This eventually led to the axiomatic definition of a vector space.



Introduction



Points in the plane R2,

points in 3-space R3,polynomials,continuous functions, anddifferentiable functions (to name only a few)






Introduction



Points in the plane R2,points in 3-space R3,

polynomials,continuous functions, anddifferentiable functions (to name only a few)






Introduction



Points in the plane R2,points in 3-space R3,polynomials,

continuous functions, anddifferentiable functions (to name only a few)






Introduction



Points in the plane R2,points in 3-space R3,polynomials,continuous functions, and

differentiable functions (to name only a few)






Introduction



Points in the plane R2,points in 3-space R3,polynomials,continuous functions, anddifferentiable functions (to name only a few)






Vector Space

Vector Space

A vector space involves four things-two sets V and F , and two algebraicoperations called vector addition and scalar multiplication.

V is a nonempty set of objects called vectors. Although V can bequite general, we will usually consider V to be a set of n-tuples or aset of matrices.

F is a scalar field-for us F is either the field R of real numbers or thefield C of complex numbers.Vector addition (denoted by x+ y) is an operation between elementsof V.Scalar multiplication (denoted by x) is an operation betweenelements of F and V.



Vector Space

Vector Space



F is a scalar field-for us F is either the field R of real numbers or thefield C of complex numbers.

Vector addition (denoted by x+ y) is an operation between elementsof V.Scalar multiplication (denoted by x) is an operation betweenelements of F and V.



Vector Space

Vector Space



F is a scalar field-for us F is either the field R of real numbers or thefield C of complex numbers.Vector addition (denoted by x+ y) is an operation between elementsof V.

Scalar multiplication (denoted by x) is an operation betweenelements of F and V.



Vector Space

Vector Space



F is a scalar field-for us F is either the field R of real numbers or thefield C of complex numbers.Vector addition (denoted by x+ y) is an operation between elementsof V.Scalar multiplication (denoted by x) is an operation betweenelements of F and V.



Cont.

Field

Let F be a subset of the real numbers R (or the complex numbers C). Weshall say that F is a field if it satisfies the following conditions:

1 If x, y are elements of F , then x+ y and xy are also elements of F .

2 If x F , then x is also an element of F . If furthermore x 6= 0 thenx1 is an element of F .

3 The elements 0 and 1 are elements of F .

For example that both R and C are fields. Verify!The set of rational numbers Q is also a field. Verify!The set of all integers Z is not a field. Why?



Cont.

Field


1 If x, y are elements of F , then x+ y and xy are also elements of F .2 If x F , then x is also an element of F . If furthermore x 6= 0 thenx1 is an element of F .





Cont.

Field




For example that both R and C are fields. Verify!

The set of rational numbers Q is also a field. Verify!The set of all integers Z is not a field. Why?



Cont.

Field




For example that both R and C are fields. Verify!The set of rational numbers Q is also a field. Verify!

The set of all integers Z is not a field. Why?



Cont.

Field







Cont.

Vector Space

The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.

Vector Addition

A1. x+ y V for all x,y V.

A2. (x+ y) + z = x+ (y + z) for everyx,y, z V

A3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that

x+ 0 = x for every x V.A5. For each x V, there exits a

(x) V such that x+ (x) = 0.

Scalar Multiplication

M1. x V for all F and x V.M2. ()x = (x) for all , F and

every x V.M3. (x+ y) = x+ y for every F

and all x,y V.M4. (+ )x = x+ x for all , F

and every x V.M5. 1x = x for every x V.



Cont.

Vector Space


Vector Addition

A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every

x,y, z V

A3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that










Cont.

Vector Space


Vector Addition


x,y, z VA3. x+ y = y + x for every x,y V.

A4. There exits a 0 V such thatx+ 0 = x for every x V.

A5. For each x V, there exits a(x) V such that x+ (x) = 0.








Cont.

Vector Space


Vector Addition


x,y, z VA3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that

x+ 0 = x for every x V.

A5. For each x V, there exits a(x) V such that x+ (x) = 0.








Cont.

Vector Space


Vector Addition












Cont.

Vector Space


Vector Addition






M1. x V for all F and x V.

M2. ()x = (x) for all , F andevery x V.

M3. (x+ y) = x+ y for every Fand all x,y V.

M4. (+ )x = x+ x for all , Fand every x V.

M5. 1x = x for every x V.



Cont.

Vector Space


Vector Addition







every x V.

M3. (x+ y) = x+ y for every Fand all x,y V.





Cont.

Vector Space


Vector Addition








and all x,y V.





Cont.

Vector Space


Vector Addition









and every x V.




Cont.

Vector Space


Vector Addition












Example 1

Because A1.A5. are generalized versions of the five additive properties ofmatrix addition, and M1.M5. are generalizations of the five scalarmultiplication properties, we can say that the following hold.

The set Rmn of m n real matrices is a vector space over R.

The set Cmn of m n complex matrices is a vector space over C.



Example 1

Because A1.A5. are generalized versions of the five additive properties ofmatrix addition, and M1.M5. are generalizations of the five scalarmultiplication properties, we can say that the following hold.

The set Rmn of m n real matrices is a vector space over R.The set Cmn of m n complex matrices is a vector space over C.



Example 2: The real coordinate spaces

The real coordinate spaces

R1n = {(x1 x2 xn), xi R} and Rn1 =

x1x2...xn

, xi R

are special cases of the preceding example, and these will be the object ofmost of our attention. In the context of vector spaces, it usually makes nodifference whether a coordinate vector is depicted as a row or as a column.Therefore we will use the common symbol Rn to designate a coordinatespace. When it is important to distinguish between rows and columns, wewill explicitly write R1n of Rn1. Similar remarks hold for complexcoordinate spaces.



Example 3

With function addition and scalar multiplication defined by

(f + g)(x) = f(x) + g(x) and (f)(x) = f(x),

the following sets are vector spaces over R:The set of functions mapping the interval [0, 1] into R.

The set of all real-valued continuous functions defined on [0, 1].The set of real-valued functions that are differentiable on [0, 1].The set of all polynomials with real coefficients.



Example 3


(f + g)(x) = f(x) + g(x) and (f)(x) = f(x),

the following sets are vector spaces over R:The set of functions mapping the interval [0, 1] into R.The set of all real-valued continuous functions defined on [0, 1].

The set of real-valued functions that are differentiable on [0, 1].The set of all polynomials with real coefficients.



Example 3


(f + g)(x) = f(x) + g(x) and (f)(x) = f(x),

the following sets are vector spaces over R:The set of functions mapping the interval [0, 1] into R.The set of all real-valued continuous functions defined on [0, 1].The set of real-valued functions that are differentiable on [0, 1].

The set of all polynomials with real coefficients.



Example 3


(f + g)(x) = f(x) + g(x) and (f)(x) = f(x),

the following sets are vector spaces over R:The set of functions mapping the interval [0, 1] into R.The set of all real-valued continuous functions defined on [0, 1].The set of real-valued functions that are differentiable on [0, 1].The set of all polynomials with real coefficients.



Example 4

Consider the vector space R2, and let

L = {(x, y)|y = x}

be a line through the origin. L is a subset of R2, but L is a special kind ofsubset because L also satisfies the properties A1.A5. and M1.M5.that define a vector space. This shows that it is possible for one vectorspace to properly contain other vector spaces.


Spaces and Subspaces Subspaces

Subspace

Subspace

Let S be a nonempty subset of a vector space V over F (symbolically,S V). If S is also a vector space over F using the same addition That is,a nonempty subset S of a vector space V is a subspace of V if and only ifA1. x,y S x+ y S, and

M1. x S x S for all F .



Subspace

Subspace

Let S be a nonempty subset of a vector space V over F (symbolically,S V). If S is also a vector space over F using the same addition That is,a nonempty subset S of a vector space V is a subspace of V if and only ifA1. x,y S x+ y S, andM1. x S x S for all F .



Example 1

Given a vector space V, the set Z = {0} containing only the zero vector isa subspace of V because A1. and M1. are trivially satisfied. Naturally, thissubspace is called the trivial subspace.



Linear Combination

Linear Combination

Let V be an arbitrary vector space, and let v1,v2, . . . ,vr be elements ofV. Let 1, 2, . . . , r be scalars. An expression of type

1v1 + 2v2 + . . .+ rvr

is called a linear combination of v1,v2, . . . ,vr.



Spanning

Spanning

For a set of vectors S = {v1,v2, . . . ,vr} from a vector space V, the set ofall possible linear combinations of the vis is denoted by

span(S) = {1v1 + 2v2 + + rvr|i F}.

Notice that span(S) is a subspace of V because the two closure propertiesA1. and M1. are satisfied. That is, if x =

i ivi and y =

i ivi are

two linear combinations from span(S), then the sumx+ y =

i(i + i)vi is also a linear combination in span(S), and for any

scalar , x =

i(i)vi is also a linear combination in span(S).



Spanning Sets

Spanning Sets

For a set of vectors S = {v1,v2, . . . ,vr} , the subspacespan(S) = {1v1 + 2v2 + + rvr} generated by forming alllinear combinations of vectors from S is called the space spanned byS.

If V is a vector space such that V = span(S), we say S is a spanningset for V. In other words, S spans V whenever each vector in V is alinear combination of vectors from S.



Spanning Sets

Spanning Sets

For a set of vectors S = {v1,v2, . . . ,vr} , the subspacespan(S) = {1v1 + 2v2 + + rvr} generated by forming alllinear combinations of vectors from S is called the space spanned byS.If V is a vector space such that V = span(S), we say S is a spanningset for V. In other words, S spans V whenever each vector in V is alinear combination of vectors from S.



Examples

S ={(

11

),

(22

)}spans the line y = x in R2.

The unit vectors

e1 =100

, e2 =010

, e3 =001

spans R3.The unit vectors {e1, e2, . . . , en} span Rn.The finite set {1, x, x2, . . . , xn} spans the space of all polynomialssuch that deg p(x) n, and the infinite set {1, x, x2, . . .} spans thespace of all polynomials.



Examples

S ={(

11

),

(22


The unit vectors

e1 =100

, e2 =010

, e3 =001

spans R3.

The unit vectors {e1, e2, . . . , en} span Rn.The finite set {1, x, x2, . . . , xn} spans the space of all polynomialssuch that deg p(x) n, and the infinite set {1, x, x2, . . .} spans thespace of all polynomials.



Examples

S ={(

11

),

(22


The unit vectors

e1 =100

, e2 =010

, e3 =001

spans R3.The unit vectors {e1, e2, . . . , en} span Rn.

The finite set {1, x, x2, . . . , xn} spans the space of all polynomialssuch that deg p(x) n, and the infinite set {1, x, x2, . . .} spans thespace of all polynomials.



Examples

S ={(

11

),

(22


The unit vectors

e1 =100

, e2 =010

, e3 =001

spans R3.The unit vectors {e1, e2, . . . , en} span Rn.The finite set {1, x, x2, . . . , xn} spans the space of all polynomialssuch that deg p(x) n, and the infinite set {1, x, x2, . . .} spans thespace of all polynomials.



Problem

For a set of vectors S = {a1,a2, . . . ,an} from a subspace V Rm1, letA be the matrix containing the ais as its columns. Explain why S spansV if and only if for each b V there corresponds a column x such thatAx = b (i.e., if and only if Ax = b is a consistent system for everyb V).


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