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^{n}to R

^{m}is actually a matrix transformation. I know that every matrix transformation is linear but not sure about the reverse.

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- Thread starter Ali Asadullah
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HallsofIvy

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^{n}to R^{m}is actually a matrix transformation. I know that every matrix transformation is linear but not sure about the reverse.

I was wondering the same when I read that bit in the textbook! To answer your question, examples of linear transformations that are not matrix transformations are those that involve non-matrix vector spaces (eg. the vector space of polynomials) and the mapping from a planar subspace of R-3 onto R-2. These examples are given later in the text too, but unfortunately David Lay does not take the trouble to point out that these linear transformations in themselves are not matrix transformations (relating them back to his earlier claim that you quote). Note however that every vector space can be coordinate-mapped onto R-n, giving each of their vectors a unique column vector representation (for example, the coordinate vector of 4+ 3t^2 relative to the standard basis for P-2 is (4,0,3)). Thus,

David Lay's textbook is

Hope this message helps any future readers of David Lay's text!

ps. Btw, do think about _why_ a linear transformation cannot be a matrix transformation when the domain/codomain is a proper subspace of R-n (i.e. does not span the whole of R-n).

pps. For completeness, let me state that each matrix can of course represent many different linear transformations.

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Stephen Tashi

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HallsofIvy

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