Hello all, I am having trouble showing linear independence at the indicated step.

The setup:

Let be a fixed vector in a finite-dimensional vector space . Let .

Prove is -invariant. (Easy)

Suppose that and show that is a basis for .

It seems obvious that checking linearly independent is easier than span, but I am getting stuck trying to get all of the coefficients to be zero. Thanks to any and all help ahead of time!