# 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: