Linear Indepence Question involving an arbitrary vector Suppose $u, v, w$ are linearly independent in $V$ and $x \in V$.
Prove $u + x, v + x, w + x$ are always linearly independent.
I'm stuck on this problem because I don't know what to do with $x$. Any hints or help would be appreciated.

You're stuck because the result is false as it is stated now. Let ${\bf x} = -{\bf u}$, or the opposite of any of the other two vectors. If $\{{\bf u},{\bf v}, {\bf w}, {\bf x}\}$ is linearly independent, for example, it seems true, and follows straight from definition.  Maybe there is some other hypothesis we can add to fix this.
