What is a vector?

Definition: A vector space, V, over a field F, is a set for which the following 10 axioms hold. For all v, w, x X V, and all l,m X F
1: v + w X V
2: (v + w) + x = v + (w + x)
3: there exist the element 0 X V such that: v + 0 = 0 + v = v
4: for all v there exists -v such that v + (-v) = 0

5: v + w = w + v

These first 5 axioms imply that the binary operation of addition +: V x V -> V forms an abelian group -
1: implies the mapping is closed (it wouldn't be a mapping if it wasn't!)
2: implies addition is associative
3: there exists an identity
4: says every element has an inverse
5: addition is commutative (the parallelogram rule) - this fith axiom is what makes the group abelian.

6: lv X V
7: ( lm) v = l( mv)
8: there exists an element 1 X F such that 1v = v for all v
9: ( l + m) v = lv + mv

10: l (v + w) = lv + lw

These last 5 axioms define the mapping from F x V -> V.

Linear dependent sets of vectors

The basis of a vector space

Let S = {v₁, v₂,...,v_n}, then S is said to span the vector space V iff every vector in V can be represented as a linear combination of element in S.
Definition: The set S = {v₁, v₂,...,v_n}, is a basis of V, iff S is a linearly independent set and S spans V.
The dimension of the vector space V is the number of elements in any basis set. Dimension is well defined because every basis of V has the same number of elements (the same cardinality).

Subspaces

Let W be a subset of V. then W is a subspace of V iff W is itself a vector space. Equivalently, W is a subspace of V iff for all u, w in W and all l,m in F:
lu + mw X W.