Can you help me out with solving the following problems. I am stuck with theses problems and I really appreciate any help.
Here is the problem 1 that I wanted to prove:
Let $M\colon H\to H$ be diagonalizable ($H$ vector space and $W$ subspace of $H$). $W$ is $M$-invariant. Prove that the restriction of $M$ to $W$ is diagonalizable and that $T\colon M/W\to M/W$ is diagonalizable (note: $M/W$ means $M$ quotient $W$ and not restriction of $M$ to $W$, i.e the quotient space).
Problem 2:
Let $M\colon H\to H$ has $n$ distinct eigenvalues (note: the dimension of $H$ is $n$). Let $U\colon H\to H$ commute with $M$. Show that $U$ is diagonalizable
For the first question, see here. If you don't know about the minimal polynomial yet, please edit your question to indicate what material you do have on hand.
For the second: Suppose that $\lambda$ is an eigenvalue of $M$, and let $\mathbf{w}$ be an eigenvector of $\mathbf{M}$ corresponding to $\lambda$. Then $\mathrm{span}(\mathbf{w})$ is the eigenspace of $\lambda$ (since $M$ has $n$ distinct eigenvalues, so each eigenvalue has algebraic and geometric dimensions equal to $1$).
Now notice that $$\lambda U(\mathbf{w}) = U\left(\lambda\mathbf{w}\right) =U(M(\mathbf{w})) = M(U(\mathbf{w})).$$ That is, if $\mathbf{z}=U(\mathbf{w})$, then $M(\mathbf{z}) = \lambda\mathbf{z}$. Thus, $U(\mathbf{w}) = \mathbf{z}\in\langle \mathbf{w}\rangle$.
Can you go from here to conclude that $U$ is diagonalizable?