There are actually several issues with your proof. The major problem is that you never actually constructed the vector . In fact, not every vector will work. For example, it is possible for the image of to be a line passing through the origin. If you had chosen to be a vector on that line, then would be linearly independent (since would simply stretch out ).

With this in mind, you need to be a little more careful. The problem may be easier to handle if you consider three cases: the nullity (i.e. dimension of the kernel) of is either 0, 1, or 2. See what you can do with each one.