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Let $V$ be a finite dimensional vector space over the field $K$, and let $W_1$ and $W_2$ be subspaces. Express $(W_1+W_2)^{\perp}$ in terms of $W_1^{\perp}$ and $W_2^{\perp}$. Also, express $(W_1\cap W_2)^{\perp}$ in terms of $W_1^{\perp}$ and $W_2^{\perp}$.

I have no idea what this exercise is asking. Remark: I am self-studying and I do not have solutions.

Questions:

What am I supposed to prove? How should I prove it?

egreg
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Pedro Gomes
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2 Answers2

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I will prove: $(W_1+W_2)^{\perp}=W_1^\perp\cap W_2^\perp.$

$W_1,W_2\subset W_1+W_2,$ and we know $(W_1+W_2)^{\perp} \subset W_1^{\perp},W_2^{\perp}$

So we get: $$(W_1+W_2)^{\perp} \subset W_1^{\perp}\cap W_2^{\perp}$$

In the other direction,

Let $v\in W_1^{\perp}\cap W_2^{\perp},$ so for every $w_1+w_2 \in W_1 + W_2:$

$$\langle v,w_1+w_2 \rangle=\langle v,w_1\rangle+\langle v,w_2\rangle=0$$

Hence, $v\in (W_1+W_2)^{\perp}.$

Itay4
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  • $\langle v,w_1\rangle$+$\langle v,w_2\rangle=0$? Why is it zero $v$ could still belong to $W_1$? – Pedro Gomes Jul 15 '17 at 16:09
  • @PedroGomes $v\in W_1^{\perp},$ how could it belong to $W_1?$ – Itay4 Jul 15 '17 at 16:10
  • Could you please expand your answer? I see $v$ can be in $W_1$ or in $W_2$, so the individual inner products would not be zero, right? – Pedro Gomes Jul 15 '17 at 16:16
  • @PedroGomes $v$ is in both $W_1^{\perp}$ and $W_2^{\perp}$, that's the definition of intersection. – Itay4 Jul 15 '17 at 16:18
  • How do you know that $(W_1+W_2)^{\perp} \subset W_1^{\perp}\cap W_2^{\perp}$ keeps the inequality $(W_1+W_2)^{\perp} \subset W_1^{\perp},W_2^{\perp}$, if $W_1^{\perp},W_2^{\perp}\subset W_1^{\perp}\cap W_2^{\perp}$? – Pedro Gomes Jul 15 '17 at 17:03
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Hint: $(W_1+W_2)^{\perp}=W_1^\perp\cap W_2^\perp$; $(W_1\bigcap W_2)^{\perp}=W_1^\perp +W_2^\perp$.

C. Ding
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    Am I supposed to prove that? How do I prove it? – Pedro Gomes Jul 14 '17 at 12:30
  • Try yourself! then you'll benefit from 'exercise'. – C. Ding Jul 14 '17 at 12:39
  • or show some effort. – C. Ding Jul 14 '17 at 12:47
  • Main thing to use in a proof (this one, and most others from elementary courses) is the definitions. Here, definition of $\perp$ and of $+$ and of $\cap$. The idea is: by thinking about this, you become familiar with the definitions. Unfortunately, reading Itay4's proof will help you much less than that. – GEdgar Jul 14 '17 at 13:10
  • @GEdgar Check out this proof of mine please. https://math.stackexchange.com/questions/2359794/w-1-bigcap-w-2-perp-w-1-perpw-2-perp-with-functionals – Pedro Gomes Jul 15 '17 at 16:41