# Change of basis

The change of basis is a technique that allows us to express vector coordinates with respect to a "new basis" that is different from the "old basis" originally employed to compute coordinates.

## Coordinates

Suppose that a finite-dimensional vector space possesses a basis .

Then, any vector can be written as a linear combination of the basis:where the scalar coefficients are uniquely determined.

Remember that the vectoris called the coordinate vector of with respect to the basis .

Example Let be a vector space and a basis for . Consider the vectorIts coordinate vector is

## The change-of-basis matrix

Suppose that we have a second basis . By the dimension theorem, and have the same number of vectors.

The first question we are going to ask is: what happens to coordinates when we switch from using as a basis to using ? In particular, how do we transform a coordinate vector into a vector of coordinates with respect to the new basis?

The answer is provided by the following proposition.

Proposition Let be a vector space. Let and be two bases for . Then, there exists a matrix, denoted by and called change-of-basis matrix, such that, for any ,where and denote the coordinate vectors of with respect to and respectively.

Proof

Let be the representation of in terms of . By the rules on the addition and scalar multiplication of coordinate vectors, we have thatwhere are the coordinate vectors of the elements of with respect to . Adjoin these vectors so as to form a matrixSincewe can write equation (1) asbecause the product is equal to a linear combination of the columns of , with coefficients taken from (see the lecture on matrix products and linear combinations). Note that does not depend on the particular choice of , as it depends only on the two bases and .

The main take-away from the previous proof is that the columns of the change-of-basis matrix are the coordinates of the vectors of the original basis with respect to the new basis :

As demonstrated by the next proposition, the change of basis matrix is invertible.

Proposition Let be a vector space. Let and be two bases for . Then, the change-of-basis matrix is invertible and its inverse equals , that is,

Proof

For any , we have thatandBy combining these two equations, we obtainThis can be true for every only if where is the identity matrix. The latter result implies that is the inverse of .

Let us make an example.

Example Consider the space of all vectors and the two bases withand withWe haveThus, the coordinate vectors of the elements of with respect to are Therefore, when we switch from to , the change-of-basis matrix isFor example, take a vectorSincethe coordinates of with respect to areIts coordinates with respect to can be easily computed thanks to the change-of-basis matrix:We can easily check that this is correct:

## Effect on the matrix of a linear operator

Remember that a linear operator on a vector space is a function such thatfor any two vectors and any two scalars and .

Given a basis for , the matrix of the linear operator with respect to is the square matrix such thatfor any vector (see also the lecture on the matrix of a linear map). In other words, if you multiply the matrix of the operator by the coordinate vector of , then you obtain the coordinate vector of .

What happens to the matrix of the operator when we switch to a new basis? The next proposition provides an answer to this question.

Proposition Let be a linear space. Let and be two bases for . Let be a linear operator. Denote by and the matrices of the linear operator with respect to and respectively. Then, or, equivalently,where and are the change-of-basis matrices that allow to switch from to and vice versa.

Proof

Let . We can use the change-of-basis matrix to transform the coordinates and Therefore, the matrix representation of the operatorcan be written asorThus, the matrixis the matrix of with respect to (which is unique). In other words,Since we can also write

Thus, the change-of-basis matrices allow to easily switch from the matrix of the linear operator with respect to the old basis to the matrix with respect to the new basis.

## Solved exercises

Below you can find some exercises with explained solutions.

### Exercise 1

Let us consider the space introduced in the example above with the two bases and . In that example, we have shown that the change-of-basis matrix isMoreover,Let be the linear operator such thatFind the matrix and then use the change-of-basis formulae to derive from .

Solution

The matrix of the linear operator with respect to the basis isThe change-of-basis formula gives