Any vector space has multiple bases, so the question naturally arises: what are the relationships between bases of a vector space? In the first place, there must be the same number of elements in any basis of a vector space. Then, given two bases of a vector space, there is a way to translate vectors in terms of one basis into terms of the other; this is known as change of basis.
Change of basis is a technique applied to finite-dimensional vector spaces in order to rewrite vectors in terms of a different set of basis elements. It is useful for many types of matrix computations in linear algebra and can be viewed as a type of linear transformation.
Suppose is a basis of and is another basis of . Then, by the definition of basis, there is precisely one way to write each in terms of 's. Specifically, there are scalars for integers such that
Then, there is an matrix that is known as the change-of-basis matrix. must be invertible because it must be injective (and it is square), by the definition of basis. Any vector could then be represented in terms of the 's by means of direct substitution:
That is, the linear transformation defined by changes the coordinates from 's to 's.
Express the vector as a linear combination of the vectors and .
This comes down to finding the scalars and so that which can be written in matrix form as . From here, we can use Gauss-Jordan elimination to conclude that and .
When using coordinates (and therefore matrices) for linear transformations, it is sometimes helpful to consider a linear transformation with a different basis.
Suppose is a linear transformation with matrix in basis and has another basis . Then, if is the change-of-basis matrix for to as above, the linear transformation has a different matrix in basis , namely
So, some vector has a representation with basis . Then, is the same vector whose representation is now with basis ; is the application of to a vector expressed in basis , yielding the output of expressed in basis ; and is the output of expressed in basis .