Find the matrix of the linear maps $L : \mathbb{R}^3\to\mathbb{R}^3$ , where $n = (n1_, n_2, n_3)$ and let $L(v) := n\times v$.

Question

Find the matrix of the linear maps $L :\mathbb{R}^3\to\mathbb{R}^3$, where $n = (n_1, n_2, n_3)$ and let $L(v) := n\times v$.

Having trouble with this question, hope someone can help.

Answer 1

Let $v = (c_1,c_2,c_3)$ be an arbitrary vector in $R^3$ then from definition of cross product $$Lv = n\times v = (n_2c_3-n_3c_2,n_3c_1-n_1c_3,n_1c_2-n_2c_1)$$ so then in particular for $v = (1,0,0),(0,1,0),(0,0,1)$ we have the following images under $L$ $$L(1,0,0) = n\times(1,0,0) = (0,n_3,-n_2)$$ $$L(0,1,0) = n\times(0,1,0) = (-n_3,0,n_1)$$ $$L(0,0,1) = n\times(0,0,1) = (n_2,-n_1,0)$$

consequently $$\mathcal{M}(L) = \begin{pmatrix}\phantom{-}0&-n_3&\phantom{-}n_2\\\phantom{-}n_3&\phantom{-}0&-n_1\\-n_2&\phantom{-}n_1&\phantom{-}0\end{pmatrix}$$