Linear Transformation problem. .

[kNYsJakE]

Homework Statement

Let \mathrm{V} be a vector space. Determine all linear transformations \mathrm{T}:V\rightarrow V such that \mathrm{T}=\mathrm{T}^2.

Homework Equations

Hint was given and it was like this:
Note that x=\mathrm{T}(x)+(x-\mathrm{T}(x)) for every x in V, and show that V=\{y:\mathrm{T}(y)=y\}\oplus\mathrm{N}(T)

The Attempt at a Solution

I tried to calculate T(x) and T^2(x) using x=T(x)+(x-T(x)) and put T and T^2 equal. But, I do not think this is how to solve this problem... =(

 

[Office_Shredder]

Each x can be written as x = T(x) + (x-T(x)) so note that:

T(T(x)) = T(x) so T(x) is in the set {y:T(y)=y}

And T(x-T(x)) = T(x) - T²(x) = T(x) - T(x) = 0 so x-T(x) is in N(T).

So given a T, we can represent V as the direct sum of the kernel of T and the image of T. The opposite question is, given V as the direct sum of two subspaces, can we find a T such that one is the kernel and the other is the image?