# Vector space

## Change of basis

where:
• : matrix in the old basis
• : matrix in the new basis
• : change of basis matrix

## Change of basis matrix

The change of basis matrix is the matrix that allows us to express the new basis in an old basis:
Mnemonic is as follows: consider we have an initial basis . Now, we define the new basis in terms of the old basis, e.g.:
which can be written in matrix form as:
and so if we set:
we have:
The usual question then is: given a vector in the new basis, how do we represent it in the old basis?
The answer is that we simply have to calculate the matrix inverse of :
That is the matrix inverse.

## Change of basis between symmetric matrices

When we have a symmetric matrix, a change of bases keeps symmetry iff it is done by an orthogonal matrix, in which case:

## Underlying field of a vector space

Every vector space is defined over a field.
E.g. in , the underlying field is , the real numbers. And in the underlying field is , the complex numbers.
Any field can be used, including finite field. But the underlying thing has to be a field, because the definitions of a vector need all field properties to hold to make sense.
Elements of the underlying field of a vector space are known as scalar.

## Scalar (mathematics)

A member of the underlying field of a vector space. E.g. in , the underlying field is , and a scalar is a member of , i.e. a real number.