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I'm revising some elementary linear algebra after a multi-decade break in which I've forgotten most of it. I've taken a look at a few introductory text books, and there seems to be a common line of argument about singular matrices that seems to me to be wrong: many of these textbooks claim that because the determinant of a singular matrix (let's say a $2 \times 2$ matrix over the reals) can be viewed as "area destroying" then the matrix maps $\mathbb{R}^2$ to $\mathbb{R}$, and it's therefore "obvious" that it cannot be inverted.

Surely this is twaddle since the cardinality of $\mathbb{R}^2$ is the same as that of $\mathbb{R}$ ? Isn't the real argument to show that a singular 2x2 matrix is not injective ?

Who's confused here? Me or the books?

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    We don't measure area (or distance, or volume) by cardinality.2011-09-22
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    While there are bijections from $\mathbb{R}$ to $\mathbb{R}^2$, it is impossible for them to be **linear**, because a basis for $\mathbb{R}$ (which has one element) would have to be in bijection with a basis for $\mathbb{R}^2$ (which has two elements). Of course, it is *also* true that a singular matrix is not injective, so one need not appeal to dimension to show that there can't be an inverse (in fact, I agree that arguing by non-injectivity is conceptually clearer). But the book is trying to give an *intuitive explanation* for why singular matrices aren't injective, not a rigorous proof.2011-09-22
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    @arturo - Sorry, I don't see what point you're making - can you expand please ? Zev: I have no problem with intuitive arguments, but it seems to me that this particular intuitive argument will leave people thinking that any map from, say, $\mathbb{R}^2$ to $\mathbb{R}$ cannot be inverted (because at this level, most people will see $\mathbb{R}^2$ as "bigger" than $\mathbb{R}$)- however, surely the main point is that lack of invertibility is due to non-injectivity, and the "area destroying" property just happens also to occur, in the case of singular matrices ?2011-09-23
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    @ukmaths: You say: "Surely [area destroying] is twaddle since the cardinality of $\mathbb{R}^2$ is the same as that of $\mathbb{R}$". But cardinality is not a measure of area; the fact that they have equal cardinality has *nothing to do* with whether or not you can consider map from $\mathbb{R}^2$ whose image is one dimensional "area destroying". The entire sentence "since the cardinality of $\mathbb{R}^2$ is the same as that of $\mathbb{R}$" is utterly irrelevant, because we don't measure area using cardinality.2011-09-23

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