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There are two proofs:

Let X and Y be two matrices such that the product XY is defined. Show that if the columns of Y are linearly dependent, then so are the columns of the matrix XY.

Let X and Y be two matrices such that the product XY is defined. Show that if the columns of the matrix XY are linearly independent, then so are the columns of Y

2. Homework Equations

N/A

3. The Attempt at a Solution

Solution for first. i do not know

Solution for second one perhaps?

If XY are assumed to be the identity matrix. Thus we know I = (A^-1)(A)

Therefore, A=X and A^-1=Y. Then we know X^-1=Y. Then by the invertible matrix theorem, the equation Ax=0 has only the trivial solution and must be linearly independent?

Those are my two cents, can anyone help me?