1Vector spaces

IB Linear Algebra

1.1 Definitions and examples

Notation. We will use F to denote an arbitrary field, usually R or C.

Intuitively, a vector space V over a field F (or an F-vector space) is a space

with two operations:

– We can add two vectors v

1

, v

2

∈ V to obtain v

1

+ v

2

∈ V .

– We can multiply a scalar λ ∈ F with a vector v ∈ V to obtain λv ∈ V .

Of course, these two operations must satisfy certain axioms before we can

call it a vector space. However, before going into these details, we first look at a

few examples of vector spaces.

Example.

(i) R

n

=

{column vectors of length n with coefficients in R}

with the usual

addition and scalar multiplication is a vector space.

An

m × n

matrix

A

with coefficients in

R

can be viewed as a linear map

from R

m

to R

n

via v 7→ Av.

This is a motivational example for vector spaces. When confused about

definitions, we can often think what the definition means in terms of

R

n

and matrices to get some intuition.

(ii)

Let

X

be a set and define

R

X

=

{f

:

X → R}

with addition (

f

+

g

)(

x

) =

f

(

x

) +

g

(

x

) and scalar multiplication (

λf

)(

x

) =

λf

(

x

). This is a vector

space.

More generally, if

V

is a vector space,

X

is a set, we can define

V

X

=

{f

:

X → V } with addition and scalar multiplication as above.

(iii) Let [a, b] ⊆ R be a closed interval, then

C([a, b], R) = {f ∈ R

[a,b]

: f is continuous}

is a vector space, with operations as above. We also have

C

∞

([a, b], R) = {f ∈ R

[a,b]

: f is infinitely differentiable}

(iv)

The set of

m × n

matrices with coefficients in

R

is a vector space, using

componentwise addition and scalar multiplication, is a vector space.

Of course, we cannot take a random set, define some random operations

called addition and scalar multiplication, and call it a vector space. These

operations have to behave sensibly.

Definition

(Vector space)

.

An

F

-vector space is an (additive) abelian group

V

together with a function F × V → V , written (λ, v) 7→ λv, such that

(i) λ(µv) = λµv for all λ, µ ∈ F, v ∈ V (associativity)

(ii) λ(u + v) = λu + λv for all λ ∈ F, u, v ∈ V (distributivity in V )

(iii) (λ + µ)v = λv + µv for all λ, µ ∈ F, v ∈ V (distributivity in F)

(iv) 1v = v for all v ∈ V (identity)

We always write

0

for the additive identity in

V

, and call this the identity.

By abuse of notation, we also write 0 for the trivial vector space {0}.

In a general vector space, there is no notion of “coordinates”, length, angle

or distance. For example, it would be difficult to assign these quantities to the

vector space of real-valued continuous functions in [a, b].

From the axioms, there are a few results we can immediately prove.

Proposition.

In any vector space

V

, 0

v

=

0

for all

v ∈ V

, and (

−

1)

v

=

−v

,

where −v is the additive inverse of v.

Proof is left as an exercise.

In mathematics, whenever we define “something”, we would also like to define

a “sub-something”. In the case of vector spaces, this is a subspace.

Definition

(Subspace)

.

If

V

is an

F

-vector space, then

U ⊆ V

is an (

F

-linear)

subspace if

(i) u, v ∈ U implies u + v ∈ U .

(ii) u ∈ U, λ ∈ F implies λu ∈ U .

(iii) 0 ∈ U.

These conditions can be expressed more concisely as “

U

is non-empty and if

λ, µ ∈ F, u, v ∈ U , then λu + µv ∈ U”.

Alternatively,

U

is a subspace of

V

if it is itself a vector space, inheriting the

operations from V .

We sometimes write U ≤ V if U is a subspace of V .

Example.

(i) {(x

1

, x

2

, x

3

) ∈ R

3

: x

1

+ x

2

+ x

3

= t} is a subspace of R

3

iff t = 0.

(ii)

Let

X

be a set. We define the support of

f

in

F

X

to be

supp

(

f

) =

{x ∈ X

:

f

(

x

)

6

= 0

}

. Then the set of functions with finite support

forms a vector subspace. This is since

supp

(

f

+

g

)

⊆ supp

(

f

)

∪ supp

(

g

),

supp(λf) = supp(f) (for λ 6= 0) and supp(0) = ∅.

If we have two subspaces

U

and

V

, there are several things we can do with

them. For example, we can take the intersection

U ∩ V

. We will shortly show

that this will be a subspace. However, taking the union will in general not

produce a vector space. Instead, we need the sum:

Definition

(Sum of subspaces)

.

Suppose

U, W

are subspaces of an

F

vector

space V . The sum of U and V is

U + W = {u + w : u ∈ U, w ∈ W }.

Proposition.

Let

U, W

be subspaces of

V

. Then

U

+

W

and

U ∩ W

are

subspaces.

Proof. Let u

i

+ w

i

∈ U + W , λ, µ ∈ F. Then

λ(u

1

+ w

1

) + µ(u

2

+ w

2

) = (λu

1

+ µu

2

) + (λw

1

+ µw

2

) ∈ U + W.

Similarly, if

v

i

∈ U ∩ W

, then

λv

1

+

µv

2

∈ U

and

λv

1

+

µv

2

∈ W

. So

λv

1

+ µv

2

∈ U ∩ W .

Both U ∩ W and U + W contain 0, and are non-empty. So done.

In addition to sub-somethings, we often have quotient-somethings as well.

Definition

(Quotient spaces)

.

Let

V

be a vector space, and

U ⊆ V

a subspace.

Then the quotient group

V/U

can be made into a vector space called the quotient

space, where scalar multiplication is given by (λ, v + U ) = (λv) + U.

This is well defined since if

v

+

U

=

w

+

U ∈ V/U

, then

v − w ∈ U

. Hence

for λ ∈ F, we have λv − λw ∈ U. So λv + U = λw + U.