Change of Basis vs. Linear Transformation

If i understand it correctly, change of basis is just a specific case of a linear transformation. Specifically given a vector space $V$ over a field $F$ such that $\dim V=n$, change of basis is just a transformation from $F^n$ to $F^n$. Does change of basis in and of itself have practical uses that are separate from linear transformations? What I mean is separate from linear transformations that do more than just change the basis of a vector in it's own vector space.

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@Asaf Karagila - Nu, I'm waiting for your answer :-) – Robert S. Barnes Jun 11 '12 at 10:27
Note: Change of bases must be invertible transformations $F^n\to F^n$. – Arturo Magidin Jun 11 '12 at 16:06
While the technical/computational aspects of a change of basis is largely given by the study of linear transformations in general, there are some philosophical distinctions that can be drawn. A nice description is given by Terry Tao on his blog/buzz. – Willie Wong Jun 12 '12 at 10:19

One way to think of a basis for $V$ is as a choice of isomorphism $F^n\to V$, where $F$ is your base field - here the basis vectors in $V$ are the image of the standard basis $(1,0,\ldots,0)$, $(0,1,\ldots,0)$ etc. in $F$. Then a change of basis is an isomorphism $F^n\to F^n$, which you precompose with the previous one to get a new isomorphism $F^n\to V$, and thus a new basis of $V$.

If you have a linear map $f\colon V\to W$, the matrix representation of $f$ depends on a choice of bases in $V$ and in $W$. Changing the bases gives you different matrix representations, and some of them are more helpful than others for, say, computing the rank, eigenvalues, determinant and so on. You don't normally have to see the change of basis map explicitly in these computations, but it's necessary for proving theorems. For example, in the row reduction algorithm, properties of the change of basis map corresponding to each row operation tell you how row operations affect the determinant.

For example, if $f\colon V\to W$ is particularly nice, you may be able to change the basis of $V$ so that the matrix of $f$ is upper triangular, meaning all entries below the diagonal are zero (here I assume $\dim{V}=\dim{W}$ so the matrix is square, but don't just write $W$ to $V$ to emphasize that we can change the basis in the domain and codomain separately). Then you can easily read off the eigenvalues - they're the entries on the diagonal - and the determinant, which is their product. If you have two composable linear maps whose matrices are simultaneously diagonalizable (i.e. you can choose bases for everything such that both matrices have all their non-zero entries on the diagonal), then you can multiply these two matrices really easily as well, just multiply the corresponding entries.

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Two questions: could you be more explicit in your examples? Also, I'm kind of emphasizing in this question practical applications of change of basis, not theoretical ones - although those are interesting. Basically, the applications of change of basis in conjunction with a linear transformation between two different vector spaces seems pretty straight forward, but what are it's practical uses in and of itself? – Robert S. Barnes Jun 11 '12 at 17:21
I've added a few more details of things you can work out from nice bases. I don't really know anything about practical uses unfortunately, hopefully you can get another answer that will say more about that. A few days ago I would have said that it's probably useful for computer algebra, but it involves inverting matrices, which turns out to be a bad idea - johndcook.com/blog/2010/01/19/dont-invert-that-matrix – Matthew Pressland Jun 12 '12 at 9:25

The Fourier transform is an example of a practical change of basis. Some often used operations in signal processing are easier in the Fourier transformed basis. For example, a convolution in the time basis is simply a multiplication in the Fourier transformed basis, i.e., the frequency basis.

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