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a) Let $W$ and $Z$ be two vector subspaces of $V$ and $W\cap Z=\{0\}$. Prove that every isomorphism from $W$ to $Z$ is the restriction of a projection operator defined on $V$.

b) Prove that if $f:k^{n}\rightarrow k^{n}$ is a linear transformation, and $\dim\,\ker f \geq \frac{n}{2}$, then $f$ is a composition of two projections.

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a) Let $f\colon W\to Z$ be an isomorphism. Then we have $f\circ\pi_1+\pi_2\colon W\oplus Z\to Z$, $(w,z)\mapsto f(w)+z$. Let $p\colon V\to W+Z=W\oplus Z$ (in the light of $W\cap Z=0$) be a projection and consider $q=(f\circ\pi_1+\pi_2)\circ p\colon V\to Z$. Then for $w \in W$ we have $q(w)=f(w)$ and for $z\in Z$ we have $q(z)=z$, in other words $f=q|_W$ and $q^2=q$ ($q$ is a projection).

b) follows readily from a)

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