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## Matrices whose range is equal to the column set [closed]

Is there such a thing?

I suppose there are two cases in which they may exist: over a finite field or infinite matrices (but let's stick to countable matrices then).

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 Zero matrices of any dimension over any field. You can construct a matrix with one element per column in a countably infinite dimensional vector space over a countable field. Why are you interested in this? – Zack Wolske May 25 2012 at 1:36

## closed as too localized by Steven Landsburg, Mark Sapir, Will Sawin, Benjamin Steinberg, Bill JohnsonMay 26 2012 at 12:57

The parity check matrix of a Hamming code, for example $$\pmatrix{1&0&0&1&1&0&1\cr0&1&0&1&0&1&1\cr0&0&1&0&1&1&1\cr}$$ almost has this property; all it's missing is an all-zero column. (This is over the field of 2 elements.)