Chapter 3

Vector Spaces

Definition and Examples

Vectors is R2

• The sum.

a

b

c

d

a+c

b+d

• Scalar multiplication.

• The length.

Area of a parallelogram

A

B

A=B

A

B

A=B

• If v =

(

a

b

)

, w =

(

c

d

)

then the area of the parallelogram

determined by v,w does not change if we add a scalar mul-

tiple of v to w.

•∣

∣

∣

∣

∣

a b

c d

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

a b

c + αa d + αb

∣

∣

∣

∣

∣

• Area of the parallelogram determined by

(

x

0

)

,

(

0y

)

is xy

and we can reduce a matrix to diagonal using row operations

of type 1.

• Area of the parallelogram determined by v,w is

∣

∣

∣

∣

∣

a b

c d

∣

∣

∣

∣

∣

Examples of vector spaces

• R, R2, Rn

• Rm×n

Axioms

Definition 1 (Real Vector Space) A set V with the opera-

tions of addition + and scalar multiplication · is called a vector

space if for every x,y ∈ V and α ∈ R, x + y ∈ V , αx ∈ V and the

following axioms are satisfied.

A1. x + y = y + x.

A2. (x + y) + z = x + (y + z).

A3. There is 0 ∈ V such that x + 0 = x.

A4. For every x ∈ V there is −x ∈ V such that x + −x = 0.

A5. For α ∈ R and x, y ∈ V , α(x + y) = αx + αy.

A6. For α, β ∈ R and x ∈ V , (α + β)x = αx + βx.

A7. For α, β ∈ R and x ∈ V , (αβ)x = α(βx).

A8. For x ∈ V , 1x = x.

Remark: Instead of R a different set of scalars can be used.

Example 1 • C[a, b].

• C over R.

• Pn, the space of polynomials of degree at most n.

• Q(√

2). The set of scalars is Q.

• The set of bounded functions f : [a, b] → R.

Subspaces

Let (V,+, ·) be a vector space and let ∅ 6= W ⊆ V . If (W,+, ·)is a vector space then it is called a subspace of V . Equivalently:

W is called a subspace of V if

• αv ∈ W for every v ∈ W and any scalar α.

• v + w ∈ W for every v,w ∈ W .

Example 2 • W = {(x1, x2, x3)T |x1 + x2 + x3 = 0} is a sub-

space of V .

• Let W be the set of functions which have a continuous nth

derivative on [0,1]. Then W is a subspace of C[0,1].

• Let W be the set of polynomials of degree at most n − 1

which are equal to 0 at x = 1. Then W is a subspace of Pn.

• Let W be the set of functions which are constant on [a, b].

Then W is a subspace of the space of bounded functions on

[a, b].

The Nullspace

Definition 2 Let A be an m×n. Then the nullspace of A, N(A)

is the subspace of Rn containing all x ∈ Rn such that Ax = 0.

The Span and the spanning set

Definition 3 Let v1, . . . ,vn ∈ V .

• Vector of the form∑n

i=1 αivi is called a linear combination

of v1, . . . ,vn.

• The span of v1, . . . ,vn, Span(v1, . . . ,vn), is the set of all linear

combinations of v1, . . . ,vn.

Theorem 1 Let v1, . . . , vn ∈ V . Then Span(v1, . . . ,vn) is a sub-

space of V .

Span(x, y)

Definition 4 Let v1, . . . ,vn ∈ V . Then the set {v1, . . . ,vn} is

called a spanning set for V if Span(v1, . . . ,vn) = V .

Remarks:

• If Span(v1, . . . ,vn) = V and v1 is a linear combination of

v2, . . . ,vn then Span(v2, . . . ,vn) = V .

• Given v1, . . . ,vn it is possible to write one of the vectors as a

linear combination of others if and only if there exist scalars

c1, . . . , cn such that

n∑

i=1

civi = 0.

Linear Independence

Definition 5 Vectors v1, . . . ,vn ∈ V are said to be linearly inde-

pendent if

n∑

i=1

civi = 0

implies that all ci = 0.

Otherwise vectors are said to be linearly dependent.

Example 3 • Show that (3,1,1)T , (1,1,0)T , (1,0,0)T are lin-

early independent in R3.

• Show that (1,1,1)T , (2,0,−2)T , (0,1,2)T are not linearly in-

dependent in R3.

• Show that f1, f2 where f1(x) = 2x + 7, f2(x) = x3 − 1 are

linearly independent in C[0,1].

• Show that 1+√

2 and 0.5 are linearly independent in Q(√

2).

• Show that ex, e−x are linearly independent in C[0,1].

Theorem 2 Let v1, . . . ,vn ∈ Rn. Then v1, . . . ,vn are linearly

independent if and only if the matrix X = (v1, . . . ,vn) is nonsin-

gular.

Theorem 3 Let v1, . . . ,vn ∈ V . Then v ∈ Span(v1, . . . ,vn) can

be written uniquely as a linear combination of v1, . . . ,vn if and

only if v1, . . . ,vn are linearly independent.

Vector Spaces of Functions

Definition 6 Let f1, . . . , fn ∈ C(n−1)[a, b]. Then W [f1, . . . , fn](x) =

det([f(i−1)j (x)]) where i, j = 1, . . . , n is called the Wronskian of

f1, . . . , fn.

Theorem 4 If f1, . . . , fn ∈ C(n−1)[a, b] are linearly dependent then

W [f1, . . . , fn](x) is the zero function.

Example 4 Let f1(x) = sin x, f2(x) = cosx and f3(x) = 2ex.

Show that f1, f2, f3 are linearly independent.

Basis and Dimension

Definition 7 The set of vectors B ⊆ V is called a basis for

V if span(B) = V and any finite subset of B contains linearly

independent vectors.

If B is finite then the above becomes: B = {v1, . . . ,vn} and

• v1, . . . ,vn are linearly independent.

• v1, . . . ,vn span V .

The standard basis for Rn

{ei|i = 1, . . . n}where

ei = (0, . . . ,0,1,0, . . . ,0).

Theorem 5 If v1, . . . ,vn span V then any set of m > n vectors

in V is linearly dependent.

Theorem 6 If v1, . . . ,vn and u1, . . . ,um are two bases for V then

m = n.

Definition 8 • If V has a finite basis then the dimension of V

is the size of basis of V and V is called finite dimensional.

• If V = {0} then dimension of V is 0.

• Otherwise V is said to be infinite-dimensional.

Theorem 7 Let V be a vector space of dimension n > 0.

• Any set of n linearly independent vectors spans V .

• Any n vectors that span V are linearly independent.

Theorem 8 Let V be a vector space of dimension n > 0.

• Any set of less than n linearly independent vectors can be

extended to a basis of V .

• Any spanning set contains a subset of n vectors that form a

basis of V .

Change of basis

Let E = [w1, . . . ,wn] and F = [v1, . . . ,vn] be two bases of V .

Problem: Given v =∑n

i=1 xiwi. Find yi so that v =∑n

i=1 yivi.

There exist sij’s such that

wi =n∑

j=1

sjivj.

and so

[w1, . . . ,wn] = [v1, . . . ,vn]S

where S = [sij] is called the transition matrix. Therefore,

S = V−1W.

If

v =n∑

i=1

xiwi

then

v =n∑

i=1

yivi

where

yi =n∑

j=1

sjixj

y = Sx.

Example 5 • Let u1 = (1,1,1)T , u2 = (1,2,2)T ,u3 = (2,3,4)T .

Find the transition matrix from [e1, e2, e3] to [u1,u2,u3].

• Find the change of basis matrix from [1, x, x2] to [1,2x,2x2−1].

Row space and Column space

Definition 9 Let A be an m × n matrix. The subspace of R1×n

spanned by rows of A is called the row space of A. The subspace

of Rm spanned by columns of A is called the column space of A.

Theorem 9 If A is row equivalent to B then the row space of

A and B are the same.

Definition 10 The rank of A is the dimension of the row space

of A.

Theorem 10 A linear system Ax = b is consistent if and only if

b is in the column space of A.

Theorem 11 An n×n matrix A is nonsingular if and only if the

column vectors of A form a basis of Rn.

Definition 11 Let A be an m × n matrix. The nullity of A is

dimension of the nullspace, N(A).

Theorem 12 (The Rank-Nullity Theorem) Let A be an m×n

matrix. Then

rank(A) + nullity(A) = n.

Theorem 13 If A is m × n then the dimension of the column

space of A and the dimension of the row space of A are the

same.