Linear Transformations

Determine a linear transformation function that would rotate the pink figure by $105^{\circ}$ clockwise and would dilate it by a factor of 0.75 . The general form of the linear transformation should be $L(x, y)=(a x+b y, c x+d y)$. You must round all coefficients to four decimal places. The resulting figure should be the green shape.

(1 point) Let $\vec{b}_{1}=\left[\begin{array}{c}-1 \\ 3\end{array}\right]$ and $\vec{b}_{2}=\left[\begin{array}{c}3 \\ -8\end{array}\right]$. The set $B=\left\{\vec{b}_{1}, \vec{b}_{2}\right\}$ is a basis for $\mathbb{R}^{2}$. Let $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a linear transformation such that $T\left(\vec{b}_{1}\right)=2 \vec{b}_{1}+6 \vec{b}_{2}$ and $T\left(\vec{b}_{2}\right)=2 \vec{b}_{1}+2 \vec{b}_{2}$. (a) The matrix of $T$ relative to the basis $B$ is (b) The matrix of $T$ relative to the standard basis $E$ for $\mathbb{R}^{2}$ is

QUESIION7 Transforming a Vector from World Space to Local Space Choose one $\cdot 5$ points A vector coordinate in world space is $\vec{r}=(1,0,2)$, calculate the vector coordinate in body frame coordinate (rigid body) $\vec{r} 0$ if the orientation of the rigid body is 90 about the global z-axis. Note: use $\overrightarrow{r_{0}}=\boldsymbol{R}^{t} \cdot \vec{r}$ $\left(\begin{array}{l}0 \\ 1 \\ 2\end{array}\right)$ $\left(\begin{array}{l}2 \\ 0 \\ 1\end{array}\right)$ $\left(\begin{array}{c}0 \\ -1 \\ 2\end{array}\right)$

Let the linear transformation \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3\) be defined by \(T(x, y, z) = (2x, 3y, -z)\). Find the pre-image of the vector \((6, 9, -3)\) under this transformation.