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Question: A Projection Transformation is defined to be a linear transformation from V to V that satisfies $T^2=T$.

Let $V=U \oplus W$ . Show that there exists only one linear projection $T:V \to V$ s.t. $KerT=W, ImT=U$

What I did

I managed to prove that if $KerT=V, ImT=${$0$} then $T^2=T$. I didn't succeed in proving that other cases are impossible (or are they possible)?

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Nontrivial projections on vector spaces certainly exist: take $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ by $[x,y]^T \mapsto [x-y, x-y]^T.$ Then $\ker T $ is precisely the line $y=x,$ a one-dimensional subspace of $\mathbb{R}^2.$

For $v\in V,$ write $v= u + w.$ This representation is unique by the definition of the direct sum operation. Since $T(w) =0,$ we have $T(v) = T(u).$ Moreover, since $T^2 (u) = T(u),$ considering the restriction map $T' = T_{|U} : U \rightarrow U = T(V)$ we have that $T'$ is a bijection - applying inverses shows that $T'(u) = u$ for all $u\in U.$ Hence $T(u+w)=u$ for all $v = u+w\in V.$ Linearity and well-definedness are easy to check.

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  • $\begingroup$ is that supposed to be u+w in the end (instead of vu+w)? $\endgroup$ – jreing May 16 '13 at 4:30

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