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# Linear algebra problem?

Topic: Homework matrix
June 21, 2019 / By Clemency
Question: Let A be and m x n matrix. A. If B is a nonsingular m x m matrix, show that BA and A have the same nullspace. B. If C is a nonsingular n x n matrix, show that AC and A have the same rank

## Best Answers: Linear algebra problem?

Bee | 10 days ago
HOMEWORK! These are easy enough that you need to see how they are done, though. A. The point is that Bx=0 if and only if x=0 since B is non-singular. Thus, BAx=0 is the same as B(Ax)=0, so Ax=0. In other words, the null space of BA is the same as that for A. B. The rank of A is the number of independent rows of the matrix A. But the k^th row of AC will be the k^th row of A multiplied by C. But since C is non-singular, it takes independent vectors to independent vectors and vice versa.
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HOMEWORK! These are easy enough that you need to see how they are done, though. A. The point is that Bx=0 if and only if x=0 since B is non-singular. Thus, BAx=0 is the same as B(Ax)=0, so Ax=0. In other words, the null space of BA is the same as that for A. B. The rank of A is the number of independent rows of the matrix A. But the k^th row of AC will be the k^th row of A multiplied by C. But since C is non-singular, it takes independent vectors to independent vectors and vice versa.