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The projection on $U$ along $W$ is the function $T:V\rightarrow V$ defined by $T(u+w)=u$, where $u \in U$, $w \in W$. Let $V=\mathbb{R}^2$, and $U=\{(x,-x): x \in \mathbb{R}\}$, and $W=\{(x,0): x\in \mathbb{R}\}$. Prove that $V=U \oplus W$, and give formulas for $T$, the projection on $U$ along $W$, and $S$, the projection on $W$ along $U$.

Is it enough to show that T is linear to show that $V=U \oplus W$? I'm uncertain as to whether that is enough to show what I want. Thanks in advance.

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    And where is the difficulty? Instead of just dumping a problem here, it's better if you show us what you've done, what you know, where you get stuck, and so on - makes it easier to write useful replies.2012-09-25
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    Hint: $(x,y) = (x+y, 0) + (-y, y)$2012-09-25
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    Sorry @GerryMyerson I typed hastily and forgot to mention that important information.2012-09-25

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