This is the problem: Suppose a, b and c are linearly independent vectors. Determine whether or not the vectors (a + b), (a - b), and (a - 2b + c) are linearly independent. Here was my solution, which involved writing words (and hasn't actually been confirmed correct yet): Let's align a, b and c with the x-y-z axis so that a and b have only x and y components, whereas c has z too. This is true for any set of linearly independent vectors. This would mean no combination of (a + b) and (a - b) can create (a - 2b + c), because neither has a z component, and therefore (a + b), (a - b), and (a - 2b + c) are also linearly independent. I'd like to know what the most efficient solution is, which I assume is algebraic and doesn't involve so many words. Thanks!