Prove that a linear transformation P : V → V of a finite dimensional vector space satisfies $P^2 = P$ if and only if there exists a basis such that the matrix of P with respect to that basis is a block matrix $$ \begin{matrix} I & 0 \\ 0 & 0 \\ \end{matrix} $$
Hence determine the minimal and characteristic polynomials of P.
I find out if P satisfies $P^2=P$, then V=ker(P) $\oplus$ img(P) But how to use this to get the block matrix and then the characteristic polynomials.
$$\lambda I-P= \begin{bmatrix} (\lambda-1)I_x & 0 \\ 0 & \lambda I_y \\ \end{bmatrix} $$ $det=(\lambda-1)^r\lambda ^y$ How to get the minimal polynomial from it?
Take a basis of the image, and a basis of the kernel, and put them together to form a basis of $V$.