The question:

Letbbe a fixed vector in Is the function defined by:

T(x) =bxxforx,

wherebxxis the cross product, a linear map? Prove your answer. Find a matrix A which transforms the vectorxinto its function T(x) = (x')

I made a complete mess of this. I tried to prove it using:

But as you can imagine, this became a massive mess very fast. Is there a particular way to prove this without resorting to performing many cross products? Thanks!