.Let L: -> be a linear operator. If A is the standard matrix representation of L, then an nxn matrix B is also a matrix representation of L.
The above doesn't make sense at all: it seems to be saying that ANY nxn matrix B is a (matrix, of course) representation of some given and fixed
linear map L, which of course is grossly false. Besides this, what is the relation between A and B??
What the question could have been, perhaps, is: prove that any nxn real matrix is the representation of some linear map from R^n to itself, but this is too
remote from the language used in the OP...
I understand that if an operator goes from two spaces that are equal (As in this case, with 2-d going to 2-d), then a matrix representation A such that L(x) = Ax for any x in R^n must be nxn. But how do I know if all nxn matrices can be standard matrix representations?
This is the way the question was written, btw.