How should I prove $My=c$ has a solution iff $c^Tv=0$ whenever $M^Tv=0$? Does it have something to do with the fundamental theorem of linear algebra?
2026-03-31 07:13:12.1774941192
Proving $My=c$ has a solution iff $c^Tv=0$ whenever $M^Tv=0$
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For one direction, multiply the equation $M y = c$ on the left by $v^T$.
For the other, you can interpret those equations as saying: whenever $v^T$ is orthogonal to the column place of M ( $v^TM = 0$), then c is orthogonal to v. This is enough to ensure that c is in the orthogonal complement to the orthogonal complement of the column space of M - from which it follows (exercise: if V is a subspace of a vector space, then [ the orthogonal complement of [ the orthogonal complement of V]] is V) that c is in the column space. This means the equation can be solved.