1. The problem statement, all variables and given/known data Problem 1: If A is an m x n matrix and Ax = 0 for all x ε ℝ^n, prove that A = O. If A and B are m x n matrices and Ax = Bx for all x ε ℝ^n, prove that A = B. (O is the 0 matrix, x is the vector x, and 0 is the 0 vector.) 2. The attempt at a solution First off, I understand the problem intuitively and can make sense of the answer being true. My issue comes in trying to phrase a proof that shows it. I know some sort of explanation of the dot product method of multiplying A by x is necessary, but I can't seem to figure out how to phrase/write/show it. Any help or advice would be much appreciated! 1. The problem statement, all variables and given/known data Problem 2: Suppose A is a symmetric matrix satisfying A^2 = O. Prove that A = O. Give an example to show that the hypothesis of symmetry is required. 2. The attempt at a solution Here I know that a symmetric matrix means that A = A^T (its transpose), also that AA = O = (A^T)A, but again I run into the issue of how it is relevant and can be turned into a proof. The problems seemed very straight forward at first glance, so I wasn't motivated to ask my professor about them as I did for others, but when I was confronted with actually writing them out, I didn't know where to begin. Any help would be wonderful, thank you!