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Given a matrix $A\in\mathbb{R}^{m\times n}$ I would like to find a basis for the null space $$ N(A) = \left\{x\in\mathbb{R}^n\,:\, Ax = 0\right\} $$ automatically.

I am looking for a general strategy that I can code into software. I would like, if possible to avoid matrix decompositions, and if possible to specifically avoid computing the SVD. Is there a matrix formula (involving inverses, if needs be) giving a set of basis vectors?

Euler_Salter
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    Let $A'$ be the row reduced form of $A$, then simply augment $(A')^T$ with the identity matrix and row reduce it again. Then you can read the basis of the image and nullspace almost directly. See here: https://math.stackexchange.com/questions/1612616/how-to-find-null-space-basis-directly-by-matrix-calculation – Igor Sep 27 '22 at 15:53
  • @Euler Would row-reduction work for your purposes, or would you prefer, for instance, a formula giving the entries of this basis in terms of determinants? – Ben Grossmann Sep 27 '22 at 18:56

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