I am beginning to learn about linear transformations and matrix-vector products, both vast subjects.
I have three questions that I have never found addressed, and I hope somebody out there can tell me or point me to the answers.
First, do there exist linear transformations in Rn that cannot be represented as matrix-vector products, where all entries are real numbers? So far as I know, there are none.
Second, do there exist matrix-vector products in Rn, where all entries are real numbers, that do not represent linear transformations? So far as I know, there are none.
Third, do there exist linear transformations in Rn, represented by matrix-vector products, that are neither isometries, dilations, nor shears? So far as I know, there are none.
I feel like I must be missing something, but I don't know what.