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