My linear algebra textbook outlines a method of finding a basis for the row space of a matrix by finding a basis for the column space of its transpose. Is there any point of using this method? It seems to me like it's adding another step (finding the transpose) for no apparent increase in simplicity.
2026-03-25 14:20:10.1774448410
Transpose method for finding a basis for the row space
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Both methods let you find a basis for the row space. The difference is that row-reducing the transpose lets you pick out a set of the original rows that form a basis, while row-reducing the original matrix gives you a “nice” basis in that the first few elements (up to the rank of the matrix) of each vector consist of a single $1$ and otherwise zeros.