# Questions on Similar matrices [ All of same type ]

1. Oct 4, 2009

### WiFO215

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 $$\neq$$ 0, Sn-1 $$\neq$$ 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 $$\neq$$ 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.

Last edited: Oct 4, 2009
2. Oct 5, 2009

### WiFO215

I don't know how this became a double post. The other thread is what I meant to post. Please delete this one.