There is an added condition of
Suppose $u_1,u_2,\ldots,u_k$ are linearly independent. Show that if A is invertible, then $Au_1,Au_2,\ldots,Au_k$ are linearly independent.
my solution
since $u_1,u_2,\ldots,u_k$ are linearly independent, then $c_1 u_1+c_2 u_2+\cdots+c_k u_k=0$ implies $c_1,\ldots$ are all zero.
by multiplying A on both sides, we have $c_1 A u_1+c_2 A u_2+\cdots =c_k A u_k=0$ since all those c can only be 0, therefore $Au_1,Au_2,\ldots,Au_k$ are linearly independent.
Notice how i didnt use the supposition if A is invertive and still get the answer. I am wondering if my answer is right or wrong...