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I remember that every embedding is injective, and every projection is surjective. For square matrices, they're both embedding and projection, which means they're bijective, so they should be invertible. But obviously, not every square matrix is invertible. I don't which part is wrong in my logic. Not every projection is surjective? Or square matrices are not embedding or projection?

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Where do you get from that a square matrix is both "embedding and projection"? Both claims are false (and, btw, what you probably meant is that square matrices are, when considered as maps, both injective (or $1-1$) and suprajective (or onto), which is also false, of course) – DonAntonio Jun 28 '12 at 3:26
@DonAntonio suprajective = surjective – user17762 Jun 28 '12 at 3:27
Thanks @Marvis. It is becoming increasingly simpler (for me, of course) to use the terms from spanish into english...:) Oh, well: you all will learn. – DonAntonio Jun 28 '12 at 3:31
@DonAntonio The tricky thing with "surjective" is that we got it from French, I think from the Bourbakistas. If it had been a more typical English word it would have been "superjective". – MJD Jun 28 '12 at 3:56
@Mark, typical English words come from French. Surplus, surcharge, surmount, surtax, .... – Gerry Myerson Jun 28 '12 at 5:28
up vote 4 down vote accepted

Consider the simplest example:

$$M: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$$

defined by the matrix $M = \begin{bmatrix} 0 & 0 \\ 0 & 0\end{bmatrix}$ is neither injective nor surjective. So square matrices needn't be injective nor surjective.

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