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I have a vector space $V$ such that $V = A\oplus A^\perp$ i.e. $V$ is a direct sum of its subspace $A$ and orthogonal complement of $A$. Suppose we also have $V = A \oplus C$

Then can we say that $A^\perp = C$. If not then in what condition this relation may hold true? I think both subspace will have same dimension but i am not sure about equality of sub spaces.

I am confused here and need a clarification.

Thanks

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    @MarianoSuárez-Alvarez Ok sir i will write.2012-05-18

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Let's rescate this question from the Unanswered Questions's limbo: $\mathbb R^2\cong Span\left\{\binom{1}{0}\right\}\oplus Span\left\{\binom{0}{1}\right\}\cong Span\left\{\binom{1}{0}\right\}\oplus Span\left\{\binom{1}{1}\right\}$and of course$\left(Span\left\{\binom{1}{0}\right\}\right)^\perp=\left(Span\left\{\binom{0}{1}\right\}\right)^\perp\neq Span\left\{\binom{1}{1}\right\}$

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    Oh, it was a typo, @talmid...I was just chatting with my brother in Mexico (I'm in Israel) and the language's neurons of my brain (all 2 of them) got confused and went into strike.2012-06-23