1. ## Linear map question

The question:

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

T(x) = b x x for x $\in \mathbb{R}^3$,

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:

$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!

2. Interesting - T is in fact linear. Do additivity and scalar multiplication seperately using the definition of cross product. The computations are quite easy.

3. Yeah, I think the method I chose to prove it was too long-winded for a cross product. It gets messy when you're calculating $\lambda_1 x + \lambda_2 y$ crossed with b.