I'm looking at a remark in Halmos' Linear Algebra Problem Book, and I feel I'm misinterpreting it.

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Here's a paraphrasing of the context:

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V is a vector space of dimension n, and A : V -> V is linear. M is a subspace of V with the property that, for any vector v in M, we have Av = v.

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We can pick a basis of M, say e_1, ..., e_m (so M has dimension m) and extend it to get a basis of V starting in e_1, ..., e_m. Then, we can write A as a matrix in a form that'll have a bunch of zeroes; particularly, as (P Q), (0 R) (first row P Q, second row 0 R), where P is an mxm matrix, 0 is a rectangular array of 0's, Q is an (n-m) x (n-m) matrix, and R is a rectangular array.

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Halmos then asks when P = 0, and discusses that it would require for the span of e_1, ..., e_m to be in the kernel of A. But wouldn't that imply that M is 0-dimensional, since it sends every vector of M to 0, yet M is invariant under A? Is he just saying this to get one to realize that, or am I missing something?