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.