Note: I am not in the course where this problem is being offered; it was simply an interesting linear algebra "thought question" that I found online to which I believe I have found a solution. However, there is one step in my solution that I am unsure about, so thank you to anyone who spares the time to assist. 1. The problem statement, all variables and given/known data Let x and y be vectors in ℝn. Is it possible that x is a scalar multiple of y (i.e., there exists a scalar c such that x = cy), but y is not a scalar multiple of x? 2. Relevant equations Basically restating the problem in an equation here, from what I see no pure equation other than this is needed: Let b, c be scalars in ℝ and x, y be vectors in ℝn. Let the scalar c be defined such that x = cy. Is there always a b such that y = bx? 3. The attempt at a solution There is not always a scalar b for the given vectors x, y and given scalar c to make the equations above true. Take the case c = 0, x = (0,0), y ≠ (0,0). Then (0,0) = 0y. But there does not exist a scalar b such that bx = b⋅(0,0) = y ≠ (0, 0) by the fact that (0,0)⋅b = (0,0) for all b. I believe I have interpreted the question correctly, and it seems that the first part of my "proof" (may not be completely formal or rigorous - I'll be open to any suggestions to improve it) is valid by the proof here. But I cannot figure out if the underlined statement is simply a postulate of linear algebra, if there is a way that I should prove it, or if it is even correct. Should the last statement of the proof be changed or is it even valid? Thanks to anyone who takes the time to help. EDIT: Clarity in last paragraph.