# Vector spaces

## Vector spaces

### Vector spaces

A vector space is a group with additional structure.

The operation for each element is shown as addition. So we can
say:

\(\forall u,v \in V [u+v \in
V]\)

To this we add scalars, from a field \(F\). We write this as multiplication.

\(\forall f \in F \forall v \in V [fv \in
V]\)

#### Subspace

A subspace is a subset of \(V\)
which still acts as a vector space. In practice, this means fewer
dimensions.

### Span

#### Span function

We can take a subset \(S\) of \(V\). We can then make linear combinations
of these elements.

This is called the linear span - \(span
(S)\).

### Linear dependence

A collection of vectors in a vector space are linearly dependent if
there exist values for \(\alpha\)
(other than all being \(0\)) such
that:

\(\sum_i \alpha_i v_i =0\).

If no such values for \(\alpha\)
exist we say the vectors are linearly independent.

### Basis vectors

#### Basis

We can write vectors as combinations of other vectors.

\(v=\sum_i \alpha_i v_i\)

A subset which spans the vector space, and which is also linearly
independent, is a basis of the vector space.

For an arbitrary vector of size \(n\), we cannot use less than \(n\) elementary vectors. We could use more,
but these would be redundant.

If we use \(n\) elementary vectors,
there is a unique solution of weights of elementary vectors.

If we use more than \(n\) elementary
vectors, there will be linear dependence, and so there will not be a
unique solution.

### Dimension function

For a basis \(S\), the the dimension
of the vector space is \(|S|\).

\(\dim (V)=|S|\)

\(S\subset V\)

#### Finite and infinite vector
spaces

If \(\dim (V)\) is finite, then we
say the vector space is finite.

Otherwise, we say the vector space is infinite.

## Points, lines and planes

### Points, lines and planes

\((1,0)\) is point, \((x,2x+1)\) is a line \((1, x, y)\) is a plane

### Parallel lines and planes

#### Parallel lines

If we have two lines:

#### Parallel planes