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Show that if $E$ is a projection on a finite dimensional vector space, then there exists a basis $B$ such that the matrix $(e_{ij})$ of $E$ with respect to $B$ has the following special form: $e_{ij}= 0$ or $1$ for all $i$ and $j$, and $e_{ij}=0$ if $i\neq j$.

I seem to be having some difficulty understand this question. I could use a little help in the right direction.

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    You could start writing in full the matrix of the projection along $\{0\}\times\{0\}\times\mathbb R$ on $\mathbb R^2\times\{0\}$ in $E=\mathbb R^3$ and ponder what this tells you on $B$ in this case.2011-10-16
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    @ Davide. I know that a projection is idempotent.2011-10-16
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    @Didier. I that that may be part of my problem. I do not know what the full matric in your example would look like.2011-10-16
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    @Rita: Why don't you work it out? Use the standard basis for $\mathbb{R}^3$; the projection sends $(a,b,c)$ to $(a,b,0)$.2011-10-16
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    @ Arturo. Thanks. The problem is, I hae only seen projections with a vector. I do not know how to work out the projection you mentioned. Do you have any suggestions?2011-10-16
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    @Rita: What is "projections with a vector"? And how is it that you know that "a projection is idempotent"? You seem to be saying on the one hand that you don't have the background to even begin to understand this question, and on the other you are using terms that suggest that you *do*.2011-10-17
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    If I may insist: what in the second paragraph of @Arturo's question (answering the request in my first comment to write in full the matrix in this specific example) is/was out of your reach? (This is a genuine question to me.)2011-10-17

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