Interesting - T is in fact linear. Do additivity and scalar multiplication seperately using the definition of cross product. The computations are quite easy.
The question:
Let b be a fixed vector in Is the function defined by:
T(x) = b x x for x ,
where b x x is the cross product, a linear map? Prove your answer. Find a matrix A which transforms the vector x into 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!