Let $\mathbf{v}_1=\begin{bmatrix}
2/3 \\ 2/3 \\ 1/3
\end{bmatrix}$ be a vector in $\R^3$.
Find an orthonormal basis for $\R^3$ containing the vector $\mathbf{v}_1$.
Add to solve later We fix a nonzero vector $\mathbf{a}$ in $\R^3$ and define a map $T:\R^3\to \R^3$ by
\[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\]
for all $\mathbf{v}\in \R^3$.
Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$.
(a) Prove that $T:\R^3\to \R^3$ is a linear transformation.
(b) Determine the eigenvalues and eigenvectors of $T$.
Add to solve later Let
\[\mathbf{v}=\begin{bmatrix}
a \\
b \\
c
\end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix}
2 \\
-1 \\
2
\end{bmatrix}.\]
Find the necessary and sufficient condition so that the vector $\mathbf{v}$ is a linear combination of the vectors $\mathbf{v}_1, \mathbf{v}_2$.
Add to solve later 
