# __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

Let S = {v_{1}, v_{2},...,v_{n}}, then S is a linearly dependent subset of V iff there exist l_{1}, l_{2},...,l_{n}, not all equal to the 0 element of F such that:

l_{1}v_{1} + l_{2}v_{2} + ... + l_{n}v_{n} = 0

S is a linearly independent set iff S is not linearly dependent.
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### The basis of a vector space

Let S = {v_{1}, v_{2},...,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 = {v_{1}, v_{2},...,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.