One can argue that a central result in our linear algebra course is that the action of any linear transformation (the action on any member of its domain) may be represented by Matrix multiplication, where the matrix's columns are obtained by "acting" with the transformation on each of the vectors in the basis of vectors that span the space in question. Furthermore, any matrix defines a unique linear transformation. 1. Please reflect on the above statement's importance (5 or so sentences). 2. The above statement implies that we have a "finite" matrix at play here, and so a finite number of basis elements in our vector space. Consider the sharp contrast of a finite dimensional mXn matrix and an infinite dimensional linear operator (transformation) like for example the ordinary derivative d/dx acting on the infinite) dimensional space of differentiable function. Can d/dx be represented by an mXn matrix; please explain under what conditions (i.e finite vs. infinite basis). One can argue that a central result in our linear algebra course is that the action of any linear transformation (the action on any member of its domain) may be represented by Matrix multiplication, where the matrix's columns are obtained by "acting" with the transformation on each of the vectors in the basis of vectors that span the space in question. Furthermore, any matrix defines a unique linear transformation. 1. Please reflect on the above statement's importance (5 or so sentences). 2. The above statement implies that we have a "finite" matrix at play here, and so a finite number of basis elements in our vector space. Consider the sharp contrast of a finite dimensional mXn matrix and an infinite dimensional linear operator (transformation) like for example the ordinary derivative d/dx acting on the infinite) dimensional space of differentiable function. Can d/dx be represented by an mXn matrix; please explain under what conditions (i.e finite vs. infinite basis)