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.
\(\forall f \in F \forall v \in V [fv \in V]\)
A subspace is a subset of \(V\) which still acts as a vector space. In practice, this means fewer dimensions.
We can take a subset \(S\) of \(V\). We can then make linear combinations of these elements.
This is called the linear span - \(span (S)\).
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.
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.
For a basis \(S\), the the dimension of the vector space is \(|S|\).
If \(\dim (V) \) is finite, then we say the vector space is finite.
Otherwise, we say the vector space is infinite.
\((1,0)\) is point, \((x,2x+1)\) is a line \((1, x, y)\) is a plane
If we have two lines: