Vector space/Change of basis/Introduction/Section

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We know, due to fact, that in a finite-dimensional vector space, any two bases have the same length, the same number of vectors. Every vector has, with respect to every basis, unique coordinates (the coefficient tuple). How do these coordinates behave when we change the bases? This is answered by the following statement.

Let be a field, and let be a -vector space of dimension . Let and denote bases of . Suppose that

with coefficients , which we collect into the -matrix

Then a vector , which has the coordinates with respect to the basis , has the coordinates

with respect to the basis .

This follows directly from

and the definition of matrix multiplication.

The matrix , which describes the base change from to , is called the transformation matrix. In the -th column of the transformation matrix, there are the coordinates of with respect to the basis . When we denote, for a vector and a basis , the corresponding coordinate tuple by , then the transformation can be quickly written as

We consider in the standard basis,

and the basis

The basis vectors of can be expressed directly with the standard basis, namely

Therefore, we get immediately

For example, the vector which has with respect to the coordinates , has the coordinates

with respect to the standard basis . The transformation matrix is more difficult to compute: We have to write the standard vectors as linear combinations of and . A direct computation (solving two linear systems) yields