1. The problem statement, all variables and given/known data 1. Let W be the space of all nx1 column matrices over R. If A is an nxn matrix over R, then A defines a linear operator La on W through left multiplication : La (X) = AX. Prove that every linear operator on W is left multiplication by some matrix A. Now, if T,S be operators such that Tn = Sn = 0 but Tn-1 [tex]\neq[/tex] 0, Sn-1 [tex]\neq[/tex] 0 . Prove that T and S both have the same matrix A for some basis B for T and B' for S. Similarly show that if M and N are nxn matrices such that Mn = Nn = 0 but Mn-1 = Nn-1 [tex]\neq[/tex] 0, then M and N are similar. 3. The attempt at a solution The first part is okay. I can always find/make some matrix A such that the column space of A is the range of La.