Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 19/Solution 1

${\mathbf {v}}={\begin{bmatrix}v_{1}\\v_{2}\\v_{3}\\\end{bmatrix}}$.

$T({\mathbf {v}})={\mathbf {v}}\times {\begin{bmatrix}1\\0\\0\\\end{bmatrix}}={\begin{vmatrix}i&j&k\\v_{1}&v_{2}&v_{3}\\1&0&0\end{vmatrix}}={\begin{bmatrix}0\\v_{3}\\-v_{2}\\\end{bmatrix}}$.

This implies that

• $T\left({\begin{bmatrix}1\\0\\0\\\end{bmatrix}}\right)={\begin{bmatrix}0\\0\\0\\\end{bmatrix}}$.

• $T\left({\begin{bmatrix}0\\1\\0\\\end{bmatrix}}\right)={\begin{bmatrix}0\\0\\-1\\\end{bmatrix}}$.
• $T\left({\begin{bmatrix}0\\0\\1\\\end{bmatrix}}\right)={\begin{bmatrix}0\\1\\0\\\end{bmatrix}}$.

That is, $T$ maps identity matrix to ${\begin{bmatrix}0&0&0\\0&0&1\\0&-1&0\end{bmatrix}}$. Thus, the matrix representation of $T$ is

$\color {blue}T={\begin{bmatrix}0&0&0\\0&0&1\\0&-1&0\end{bmatrix}}$.