Again, no. In fact, matrices are themselves linear transformations. Since any n-dimensional vector space is isomorphic to Rn, any given n by m matrix represents a linear transformation from a m-dimensional vector space to an n-dimensional vector space.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.
Again, no. Every linear transformation is a combination of those three kinds of linear transformations.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.