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I came across a statement that:

if A and B are matrices of the same order, and the rank of B = the rank of AB, then the solution space of the homogenous system $AB \underline{x}=0$ is identical to the solution space to the homogenous system $B \underline{x}=0$.

No proof is provided, and I am a bit puzzled at why this would be the case? Can anyone provide some insight?

Thank you!

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The rows of $AB$ are linear combinations of the rows of $B$. If the ranks of $B,AB$ are identical, the rowspace of $B$ and $AB$ is the same, and the nullspace of $B,AB$ is also the same, being the orthogonal complement of the row space.

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In general, $x$ will be a solution to $ABx = 0$ if either $Bx = 0$, or $Bx \neq 0$ and $ABx = 0$. Consequently, every solution to $Bx = 0$ is also a solution to $ABx = 0$.

If we are given than $AB$ and $B$ have the same rank, however, then we can conclude (by the rank-nullity theorem) that the nullspaces of $AB$ and $B$ have the same dimension. So, the solution space to $Bx = 0$ is a subspace of $ABx$, but both subspaces have the same dimension. So, the subspaces (i.e. the solutions spaces) must be identical.

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  • $\begingroup$ I'm not sure the first line is correct. Did you mean $ABx=0$? $\endgroup$ – Shubham Johri Jan 8 at 19:31
  • $\begingroup$ @ShubhamJohri yes, good catch $\endgroup$ – Omnomnomnom Jan 8 at 19:35

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