Linear Transformation problem. Please Help.

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Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

Answers and Replies

  • #2
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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?