- : matrix in the old basis
- : matrix in the new basis
- : change of basis matrix
The change of basis matrix is the matrix that allows us to express the new basis in an old basis:
The usual question then is: given a vector in the new basis, how do we represent it in the old basis?
That is the matrix inverse.
Every vector space is defined over a field.
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.