Linear Algebra: Linear Transformations(1)

Hello,

I am teaching myself Linear Algebra following P.R.Halmos' "Finite Dimensional Vector Spaces".

I am stuck at a couple of problems:

-----------------------------------------------------------------------------------------------------------------------------

** **

Problem (1)

Prove that corresponding to every linear transformation A on a finite dimensional vector space V, there exists an invertible linear transformation P such that APA = A.(Or equivalently prove that PA is a projection)

-----------------------------------------------------------------------------------------------------------------------------

**Problem (2)**

If A,B,C are linear transformations on a finite dimensional vector space, then does (AB - BA)^2 commutes with C always?

What happens when the dimension of the vector space is 2?

-----------------------------------------------------------------------------------------------------------------------------

**Problem (3)**

From the basic definition of a determinant of a linear transformation(see below), prove that $\displaystyle \text{det }(A \otimes B) = \text{det }(A) ^m \text{det }(B) ^n$, where A and B are linear transformations on vector spaces of dimensions n and m respectively

(I know the proof of a particular case when A and B are matrices, using eigenvalues. But I would love to see a proof using the below definition).

-----------------------------------------------------------------------------------------------------------------------------

__Definition of Determinant of a linear transformation__:

If W is the space of all alternating n-linear forms on an n-dimensional vector space V,then we know that it is one-dimensional. Thus given a linear transformation A, we have a scalar $\displaystyle \delta$ such that for every w in W, $\displaystyle w(Ax_1,Ax_2,.....,Ax_n) = \delta w(x_1,x_2,....,x_n)$$\displaystyle \forall x_1,x_2,...x_n$ vectors in V. Then the det(A) is defined to be the scalar $\displaystyle \delta$.

Thank you,

Srikanth