The question:

Letbbe a fixed vector in $\displaystyle \mathbb{R}^3.$ Is the function $\displaystyle T:\mathbb{R}^3 -> \mathbb{R}^3$ defined by:

T(x) =bxxforx$\displaystyle \in \mathbb{R}^3$,

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:

$\displaystyle T(\lambda_1 x + \lambda_2 y) = \lambda_1 T(x) + \lambda_2 T(y)

$

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!