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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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    @Gerry I have not counted carefully, but I think "super-" words outnumber "sur-" words by about five to one.2012-06-28

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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.