No. Every linear transfromation from Rn to Rn can be expressed as an n by n matrix. The exact matrix depends upon your choice of basis for Rn- though different matrices represent the same linear transformation in different bases if and only if they are "similar matrices".

Again, no. In fact, matricesSecond, 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.arethemselves 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.

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.

Thanks!