Linear transformation and matrix transformation

[Ali Asadullah]

Do all linear transformations are matrix transformation? In a book by David C Lay, he wrote on page 77 that not all linear tranformations are matrix transformations and on page 82 he wrote that very linear transformation from Rⁿ to R^m is actually a matrix transformation. I know that every matrix transformation is linear but not sure about the reverse.

 

[HallsofIvy]

Every linear transformation can be represented by a matrix multiplication. But writing a linear transformation as a matrix requires selecting a specific basis. If you are talking about $R^n$ to $R^m$ (there are other vector spaces) and are using the "standard" basis, then, yes, you can identify any linear transformation with a specific matrix and vice-versa.

 

[Fredrik]

See this post for more about the connection between linear operators and matrices.

 

[worldbook]

Do all linear transformations are matrix transformation? In a book by David C Lay, he wrote on page 77 that not all linear tranformations are matrix transformations and on page 82 he wrote that very linear transformation from Rⁿ 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, every linear transformation from a vector space V has a unique matrix representation after all! (The matrix acts on the coordinate vectors of the vectors in V, not the vectors in V themselves.)

David Lay's textbook is horrible as a reference text because the material is all over the place (especially on linear transformations), but he has some valid pedagogical reasons for structuring the book the way he does. He shouldn't have made that claim though, as it's an unimportant technicality at that point in the text that causes unnecessary confusion yet becomes patently self-evident later on.

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.

 

[Stephen Tashi]

There are infinite dimensional vector spaces. Is the definition of linear transformation being discussed in this thread restricted to a mapping from one finite dimensional vector space to another? (I assume the definition of "matrix" that is being discussed refers to a finite dimensional array.)

 

[HallsofIvy]

Certainly, when I said "any linear transformation can be represented as a matrix", I was thinking of the finite dimensional case. Thanks for clarifying that.