Dot product and cross product as linear transformations

#linear_algebra

Let $R^{3}$ denote the $3$ -dimensional vector space, and let $v = (a, b, c)$ be a fixed nonzero vector. The maps $C : R^{3} \to R^{3}$ and $D : R^{3} \to R$ defined by $C (w) = v \times w$ and $D (w) = (v \cdot w) v$ are linear transformations.

Determine the eigenvalues of $C$ and $D$ .
Determine the eigenspaces of $C$ and $D$ as subspaces of $R^{3}$ , in terms of $a, b, c$ .
Find a matrix for $C$ with respect to the standard basis.

Show all work and explain reasoning.

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Let ${\bf R}^3$ denote the $3$-dimensional vector space, and let ${\bf v}=(a,b,c)$ be a fixed nonzero vector. The maps $C:{\bf R}^3\to {\bf R}^3$ and $D:{\bf R}^3\to {\bf R}$ defined by $C({\bf w})={\bf v}\times {\bf w}$ and $D({\bf w})=({\bf v}\cdot {\bf w}){\bf v}$ are linear transformations.
	\begin{enumerate}[label=\alph*)]
		\item Determine the eigenvalues of $C$ and $D$.
		
		\item Determine the eigenspaces of $C$ and $D$ as subspaces of ${\bf R}^3$, in terms of $a, b, c$.
		
		\item Find a matrix for $C$ with respect to the standard basis. 
	\end{enumerate}
Show all work and explain reasoning.