Chapter 3

Vector Spaces

The operations of addition and scalar multiplication

are used in many contexts in mathematics. Regardless

of the context, however, these operations usually obey

the same set of algebra rules. Thus a general theory of

mathematical systems involving addition and scalar

multiplication will have application to many areas in

mathematics.

Mathematical systems of this form are called

vector spaces or linear spaces.

1 Definition and Examples

Euclidean Vector Spaces Rn

In general, scalar multiplication and addition in Rn are

defined by

nn yx

yx

yx

yx

22

11

nx

x

x

x

2

1

and

for any nRyx , and any scalar 。

The Vector Space Rm×n

Rm×n denote the set of all m×n matrices with real entries.

Definition

Let V be a set on which the operations of addition and

scalar multiplication are defined. By this we mean that,

with each pair of elements x and y in V, we can associate

a unique elements x+y that is also in V, and with each

element x in V and each scalar , we can associate a

unique element x in V. The set V together with the

operations of addition and scalar multiplication is said

to form a vector space if the following axioms are satisfied.

Vector Space Axioms

A1. x+y=y+x for any x and y in V.

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

A3. There exists an element 0 in V such that x+0=x for

each x ∈V.

A4. For each x ∈V, there exists an element –x in V such

that x+(-x)=0.

A5. α(x+y)= αx+ αy for each scalar α and any x and y in V.

A6. (α+β)x=αx+βx for any scalars α and β and any x ∈V.

A7. (αβ)x=α(βx) for any scalars α and β and any x ∈V.

A8. 1·x=x for all x ∈V.

The closure properties of the two operations:

C1. If x V and ∈ α is a scalar, then αx V .∈C2. If x,y V∈ , then x+y V∈ .

Example Let W={(a,1) a real} with addition and scalar

Multiplication defined in the usual way.

Example Let S be the set of all ordered pairs of real

numbers. Define scalar multiplication and addition on

S by

)0,(),(),(

),(),(

112121

2121

yxyyxx

xxxx

We use the symbol to denote the addition operation

for this system avoid confusion with the usual addition

x+y of row vectors. Show that S, with the ordinary scalar

multiplication and addition operation ,is not a vector

space. Which of the eight axioms fail to hold?

The Vector Space C[a, b]

C[a,b] denote the set of all real-valued functions that

are defined and continuous on the closed interval

[a,b].

)())((

)()())((

xfxf

xgxfxgf

The Vector Space Pn

Pn denote the set of all polynomials of degree less

than n.

)())((

)()())((

xpxp

xqxpxqp

Theorem 3.1.1 If V is a vector space anf x is any

Element of V, then

(1) 0x=0.

(2) x+y=0 implies that y=-x.

(3) (-1)x=-x.

2 Subspaces

Example Let , S is a subset of R2.

12

2

1 2xxx

xS

Definition

If S is a nonempty subset of a vector space V, and S

satisfies the following conditions:

(1) whenever for any scalar

(2) whenever and

then S is said to be a subspace of V.

Sx Sx

Syx Sx Sy

Example

1、 Let , S is a subspace of R3. 21321 xxxxxS T

2、 Let . If either of the two

conditions in the definition fails to hold, then S will not

be a subspace.

number real a is

1x

xS

3、 Let .The set S forms a subspace

of R2×2 .

211222 aaRAS

The Nullspace of a Matrix

Let A be an m×n matrix. Let N(A) denote the set of all

solutions to the homogeneous system Ax=0. Thus

0)( AxRxAN n

The subspace N(A) is called the nullspace of A.

Example Determine N(A) if

1012

0111A

The Span of a Set of Vectors

Definition

Let v1, v2, … , vn be vectors in a vector space V. A sum of the f

orm α 1v1+ α 2v2+ ‥‥ α nvn, where α1, …, αn are scalars, is calle

d a linear combination of v1, v2, … , vn .

The set of all linear combinations of v1, v2, … , vn is called th

e span of v1, v2, … , vn . The span of v1, v2, … , vn will be den

oted by Span(v1, …, vn).

Theorem 3.2.1 If v1, v2, … , vn are elements of a vector

space V, then Span(v1, v2, … , vn ) is a subspace of V.

Spanning Set for a Vector Space

Definition

The set {v1, v2, … , vn} is a spanning set for V if and only if eve

ry vector in V can be written as a linear combination of v1, v2,

‥‥ ,vn.

Example Which of the following are spanning sets for R3?

1 2 3(a) , , , 1 2 3

(b) 1 1 1 , 1 1 0 , 1 0 0

(c) 1 0 1 , 0 1 0

(d) 1 2 4 , 2 1 3 , 4 1 1

T

T T T

T T

T T T

e e e

3 Linear Independence

Consider the following vectors in R3:

8

3

1

x,

1

3

2

x,

2

1

1

x 321

Conclusion:

(1) If v1, v2, … , vn span a vector space V and one of these

vectors can be written as a linear combination of the other

n-1 vectors, then those n-1 vectors span V.

(2) Given n vectors v1, v2, … , vn , it is possible to write one of

the vectors as a linear combination of the other n-1 vectors if

and only if there exist scalars c1, …, cn not all zero such that

c1v1+c2v2+‥‥cnvn=0

Definition

The vectors v1, v2, … , vn in a vector space V are said to be li

nearly independent if

c1v1+c2v2+‥‥+cnvn=0

implies that all the scalars c1, …, cn must equal 0.

Definition

The vectors v1, v2, … , vn in a vector space V are said to be li

nearly dependent if there exist scalars c1, …, cn not all zero

such that

c1v1+c2v2+‥‥+cnvn=0

If there are nontrivial choices of scalars for which the linear

combination c1v1+c2v2+‥‥+cnvn equals the zero vector, then

v1, v2, … , vn are linearly dependent. If the only way the linear

combination c1v1+c2v2+ +‥‥ cnvn can equal the zero vector is

for all the scalars c1, …, cn to be 0, then v1, v2, … , vn are line

arly independent.

Theorems and Examples

Example Which of the following collections of vectors are linearly independent in R3?

(a) 1 1 1 , 1 1 0 , 1 0 0

(b) 1 0 1 , 0 1 0

(c) 1 2 4 , 2 1 3 , 4 1 1

T T T

T T

T T T

Theorem 3.3.1 If x1, x2, … , xn be n vectors in Rn and let

X=(x1, …, xn). The vectors x1,x2, …, xn will be linearly

dependent if and only if X is singular.

Example Determine whether the vectors (4, 2, 3)T,

(2, 3, 1)T, and (2, -5, 3)T are linearly dependent.

Example Given

7

7

0

1

x,

2

1

3

2

x,

3

2

1

1

x 321

determine if the vectors are linearly independent.

Theorem 3.3.2 If v1, v2, … , vn be vectors in a vector

space V. A vector v in Span(v1, …, vn) can be written

uniquely as a linear combination of v1,v2, …, vn if and only

if v1,v2, …, vn are linearly independent.

4 Basis and DimensionDefinition

The vectors v1, v2, … , vn form a basis for a vector space V if

and only if

(1) v1, …, vn are linearly independent.

(2) v1, …, vn span V.

Example The standard basis for R3 is {e1, e2, e3};however,

there are many bases that we could choose for R3.

Example In R2×2, consider the set {E11, E12, E21, E22},

where

10

00,

01

00

00

10,

00

01

2221

1211

EE

EE

Theorem 3.4.1 If {v1, v2, … , vn } is a spanning set for a

vector space V, then any collection of m vectors in V,

where m>n, is linearly dependent.

Corollary 3.4.2 If {v1, v2, … , vn } and {u1, u2, … , um } are

both bases for a vector space V, then n=m.

Definition

Let V be a vector space. If V has a basis consisting of n v

ectors, we say that V has dimension n. The subspace {0} of

V is said to have dimension 0. V is said to be finite-dimensio

nal if there is a finite set of vectors that spans V; otherwise,

we say that V is infinite-dimensional.

Theorem 3.4.3 If V is a vector space of dimension n>0:

(1) Any set of n linearly independent vectors spans V.

(2) Any n vectors that span V are linearly independent.

Example Show that is a basis

for R3.

1

0

1

0

1

2

3

2

1

，，

Theorem 3.4.4 If V is a vector space of dimension

n>0,

then:

(1) No set of less than n vectors can span V.

(2) Any subset of less than n linearly independent

vectors can be extended to form a basis for V.

(3) Any spanning set containing more than n vectors

can be pared down to form a basis for V.

5 Change of Basis

Changing Coordinates in R2

x=x1e1+x2e2 the coordinate of x is (x1, x2)T

x=αy+βz the coordinate of x is (α, β)T

Changing Coordinates

Let [e1, e2] be the standard basis, [u1, u2] is another basis.

Two problems:

(1) Given a vector x=(x1, x2)T, find its coordinates with

respect to u1 and u2.

(2) Given a vector c1u1+c2u2, find its coordinates with

respect to e1 and e2.

1

1u

2

3u 21 ，

x=Uc

the matrix U is called the transition matrix from the ordered

basis [u1, u2] to the basis [e1, e2].

Example Let u1=(3,2)T, u2=(1,1)T, and x=(7,4)T. Find

the coordinates of x with respect to u1 and u2.

Example Let b1=(1,-1)T, b2=(-2,3)T. Find the transition

matrix from [e1, e2] to [b1, b2] and determine the coordinates

of x=(1,2)T with respect to [b1, b2].

Example Find the transition matrix corresponding to the

change of basis from [v1, v2] to [u1, u2], where

3

7v,

2

5v 21 and

1

1u

2

3u 21 ，

Change of Basis for a General Vector Space

Definition

Let V be a vector space and let E=[v1, v2, … , vn] be an ord

ered basis for V. If v is any element of V, then v can be writte

n in the form

v=c1v1+c2v2+‥‥+cnvn

where c1, …, cn are scalars. Thus we can associate with each

vector v a unique vector c=(c1, c2, …, cn)T in Rn. The vector

c defined in this way is called the coordinate vector of v with

respect to the ordered basis E and is denoted [v]E. The ci’s ar

e called the coordinates of v relative to E.

Example Let

E=[v1, v2, v3]=[(1, 1, 1)T, (2, 3, 2)T, (1, 5, 4)T]

F=[u1, u2, u3]=[(1, 1, 0)T, (1, 2, 0)T, (1, 2, 1)T]

Find the transition matrix from E to F. If

x=3v1+2v2-v3 and y=v1-3v2+2v3

find the coordinates of x and y with respect to the ordered

basis F.

6 Row Space and Column Space

Definition

If A is an m×n matrix, the subspace of R1×n spanned by the

row vectors of A is called the row space of A. The subspace

of Rm spanned by the column vectors of A is called the column

space of A.

Theorem 3.6.1 Two row equivalent matrices have the same

row space.

Definition

The rank of a matrix A is the dimension of the row space of A.

Example Let

741

152

321

A

Theorem 3.6.2 (Consistency Theorem for Linear Systems)

A linear system Ax=b is consistent if and only if b is in the

column space of A.

Theorem 3.6.3 Let A be an m×n matrix. The linear system

Ax=b is consistent for every b∈Rm if and only if the column

vectors of A span Rm. The system Ax=b has at most one

solution for every b∈Rm if and only if the column vectors of A

are linearly independent.

Corollary 3.6.4 An n×n matrix A is nonsingular if and only if

the column vectors of A form a basis for Rn.

Definition

The dimension of the nullspace of a matrix is called the nullity

of the matrix.

Theorem 3.6.5 ( The Rank-Nullity Theorem)

If A is an m×n matrix, then the rank of A plus the nullity of A

equals n.

Example Let

5121

0342

1121

A

Find a basis for the row space of A and a basis for N(A).

Verify that dim N(A)=n-r.

Theorem 3.6.6 If A is an m×n matrix, the dimension of the

row space of A equals the dimension of the column space of A.

Example Let

513521

43110

22031

21121

A

Find a basis for the column space of A.

Example Find the dimension of the subspace of R4

spanned by

4

5

8

3

x,

0

2

4

2

x,

2

3

5

2

x,

0

1

2

1

x 4321