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.

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.
