Linear Transformation Question

Quote:

Originally Posted by flaming

Let v be a fixed vector in $R^3$ . Show that the transformation defined by T(u) = v x u is a linear transformation.

I am confused because is this a special case where V = W(by definition), so the linear transformation is a linear operator?

If that is the case how do i show this is a linear transformation?

You can try to show that this is a linear transformation in one step by checking if $T\left(\alpha u_1+\beta u_2\right)=\alpha T\left(u_1\right)+\beta T\left(u_2\right)$ , where $u_1,u_2\in\mathbb{R}^3$ :

$T\left(\alpha u_1+\beta u_2\right)=\left(\alpha u_1+\beta u_2\right)\times v$

Now from cross product properties:

$\left(\mathbf{u}+\mathbf{v}\right)\times\mathbf{w} =\mathbf{u}\times\mathbf{w}+\mathbf{v}\times\mathb f{w}$ and $\left(k\mathbf{u}\right)\times\mathbf{v}=k\left(\m athbf{u}\times\mathbf{v}\right)$

Thus,

$\begin{aligned}T\left(\alpha u_1+\beta u_2\right)&=\left(\alpha u_1+\beta u_2\right)\times v\\&=\left(\alpha u_1\right)\times v+\left(\beta u_2\right)\times v\\&=\alpha\left(u_1\times v\right)+\beta \left(u_2\times v\right)\\&=\alpha T\left(u_1\right)+\beta T\left(u_2\right)\end{aligned}$

Thus, $T$ is a linear transformation.

Does this make sense?