91Ó°ÊÓ

Find the matrix for the linear transformation which reflects every vector in \(\mathbb{R}^{2}\) across the \(x\) axis and then rotates every vector through an angle of \(\pi / 6 .\)

Find the matrix for \(T(\vec{w})=\operatorname{proj}_{\vec{v}}(\vec{w})\) where \(\vec{v}=\left[\begin{array}{ccc}1 & 5 & 3\end{array}\right]^{T}\).

Find the matrix for the linear transformation which reflects every vector in \(\mathbb{R}^{2}\) across the y axis and then rotates every vector through an angle of \(\pi / 6 .\)

Find the matrix for \(T(\vec{w})=\operatorname{proj}_{\vec{v}}(\vec{w})\) where \(\vec{v}=\left[\begin{array}{lll}1 & 0 & 3\end{array}\right]^{T}\).

Write the solution set of the following system as a linear combination of vectors $$ \left[\begin{array}{llll} 1 & 1 & 0 & 1 \\ 1 & -1 & 1 & 0 \\ 3 & 1 & 1 & 2 \\ 3 & 3 & 0 & 3 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \\ w \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] $$

Find the matrix for the linear transformation which rotates every vector in \(\mathbb{R}^{2}\) through an angle of \(5 \pi / 12 .\) Hint: Note that \(5 \pi / 12=2 \pi / 3-\pi / 4\).

Find the matrix of the linear transformation which rotates every vector in \(\mathbb{R}^{3}\) counter clockwise about the z axis when viewed from the positive z axis through an angle of \(30^{\circ}\) and then reflects through the xy plane.

Write the solution set of the following system as a linear combination of vectors $$ \left[\begin{array}{llll} 1 & 1 & 0 & 1 \\ 2 & 1 & 1 & 2 \\ 1 & 0 & 1 & 1 \\ 0 & -1 & 1 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \\ w \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] $$

Suppose \(A \vec{x}=\vec{b}\) has a solution. Explain why the solution is unique precisely when \(A \vec{x}=\overrightarrow{0}\) has only the trivial solution.