Introduction to Vector Spaces

[Notation: $\exists =$ ``there exists'', $\forall =$ ``for all'', ${\mathbbR} =$ ``the reals,'' i.e. $(-\infty,\infty)$]

A vector space, V, is a set together with two operations, vector addition and scalar multiplication, that satisfies the following properties:

1.

Closure under addition:

If $u,v \in V$, then $u + v \in V$.

2.

Commutativity of vector addition:

u + v = v + u

3.

Associativity of vector addition:

(u + v) + w = u + (v + w)

4.

Existence of an additive identity

$\exists \: {\bf 0} \in V, \textrm{such that} \: u + {\bf 0} = u, \forall u \in V$ (the zero vector)

5.

Existence of additive inverses

$\forall u \in V, \exists v \in V, \textrm{such that}\: u + v = {\bf 0}.$
We generally denote such a v as -u.

6.

Closure under scalar multiplication

If $v \in V$, then $cv \in V$, for all $c \in {\mathbb R}$.

7.

Distributive property 1

c(u + v) = cu + cv

8.

Distributive property 2

(c + d)u = cu + du

9.

Associative property for scalar multiplication

If $u \in V$, then c(du) = (cd)u, $\forall c,d \in {\mathbb R}$.

10.

Scalar identity

$\forall u \in V, 1(u) = u.$

A set of vectors, S = {v₁, v₂, ..., v_n} is linearly independent if the only solution (in the c_i's) to the equation

$$c_1v_1 + c_2v_2 + ... + c_nv_n = {\bf 0}$$

occurs with c₁ = c₂ = ... = c_n = 0. If there is some solution where some of the c_i's are nonzero, then the set S is linearly dependent.

A basis is a set, B, of vectors that satisfies the following properties:

1.

B is linearly independent

2.

$\forall v \in V$, $B \cup \{v\}$ is linearly dependent.

Let V be a vector space. If B₁ and B₂ are bases for V, then |B₁| = |B₂| (B₁ and B₂ have the same size).

The dimension of a vector space V is the size of a set of basis vectors. If B is a basis for V, we write $\dim{V} = \vert B\vert$.