Proving solution set equivalency

Question

If a 7 x 9 matrix is A and a 7 x 7 invertible matrix is B, how do I prove that the solution set for system of linear equations Ax=0 is the same for the solution set system of linear equations (BA)x=0? I have learned that Ax=0 is a homogenous linear system and that all the solutions are 0; and that for a square matrix to be invertible is has to be one to one and onto. However, where does one get started and complete the proof? Can I just prove it through an example? Though this seems tedious with such large matrices

Answer 1

Let $S = \{x \mid Ax = 0\}$ and let $T = \{x \mid (BA)x = 0\}$. We want to show that $S = T$. To do this, we show that each set contains the other:

• $\boxed{\subseteq}:$ Choose any $x \in S$ so that $Ax = 0$. How can we manipulate this equation to conclude that $(BA)x = 0$ so that $x \in T$?

• $\boxed{\supseteq}:$ Choose any $x \in T$ so that $(BA)x = 0$. How can we manipulate this equation to conclude that $Ax = 0$ so that $x \in S$?