But the fact that the "algorithm" for checking infectivity for a linear map given by a matrix is to check that all cols are pivotal, and for surjectivity that all rows are pivotal, seems to convey the same(-ish) idea. 2/2
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Of course, all this would be clearer if we were allowed to talk about dual spaces. But alas, while we can talk about the transpose of a matrix, we cannot interpret it as the matrix of the transpose of the original map.
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