How to prove that the cross product of two vectors is a linear transformation?

Question

I'm stuck, could I please have a hint?

Answer 1

(tacitly assuming that we are working with the traditional cross product of two vectors in $\mathbb{R}^3$)

Assuming that one of the vectors is held constant. Let $v$ be the fixed vector. We wish to prove that $L(x) = v\times x$ is a linear transformation of $x$ from $\mathbb{R}^3$ to $\mathbb{R}^3$.

Let $v = \begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}$. Let $x = \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$.

The cross product $v\times x = \det\left(\begin{bmatrix}\hat{i}&\hat{j}&\hat{k}\\ v_1&v_2&v_3\\x_1&x_2&x_3\end{bmatrix}\right) = \begin{bmatrix}v_2x_3-v_3x_2\\-v_1x_3+v_3x_1\\v_1x_2-v_2x_1\end{bmatrix} = x_1\begin{bmatrix}0\\v_3\\-v_2\end{bmatrix}+x_2\begin{bmatrix}-v_3\\0\\v_1\end{bmatrix}+x_3\begin{bmatrix}\star\\\star\\\star\end{bmatrix}$

Letting you fill in the $\star$'s above yourself.

How can we construct a matrix then such that $Ax = v\times x$?

If we successfully construct such a matrix, then that shows that the transformation is indeed a linear transformation.

Answer 2

For a function $L: \mathbb{R}^3 \to \mathbb{R}^3$ to be a linear transformation, you have to verify two properties:

1. $L(a \overrightarrow{v})=aL(\overrightarrow{v})$ for any scalar $a$ and any vector $\overrightarrow{v}$
2. $L(\overrightarrow{v}+\overrightarrow{w})=L(\overrightarrow{v})+L(\overrightarrow{w})$ for any two vectors $\overrightarrow{v}$ and $\overrightarrow{w}$

In this case, fix some vector $\overrightarrow{u}$, and consider the linear transformation defined by $L(\overrightarrow{v})= \overrightarrow{u} \times \overrightarrow{v}$. To show that the cross-product is linear, you need to show that properties (1) and (2) above hold; in other words, you need to verify that:

1. $\overrightarrow{u} \times (a\overrightarrow{v})=a ( \overrightarrow{u} \times \overrightarrow{v})$
2. $\overrightarrow{u} \times (\overrightarrow{v} + \overrightarrow{w})=(\overrightarrow{u}\times \overrightarrow{v})+(\overrightarrow{u}\times \overrightarrow{w})$

Can you take it from there?