Linear Transformation

Let $T$ be defined by $T(\vec{x})=A \vec{x}$. Find a vector $\vec{x}$ whose image under $T$ is $\vec{b},$ and determine whether $\vec{x}$ is unique.   solution

$A=\left[\begin{array}{rrr}{1} & {0} & {-2} \\ {-2} & {1} & {6} \\ {3} & {-2} & {-5}\end{array}\right], \quad \vec{b}=\left[\begin{array}{r}{-1} \\ {7} \\ {-3}\end{array}\right]$

Let $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ be a linear transformation such that $T\left(x_{1}, x_{2}\right)=\left(x_{1}-2 x_{2}, -x_{1}+3 x_{2},  3 x_{1}-2 x_{2}\right)$. Find $\vec{x}$ such that $T(\vec{x})=(-1,4,9)$.  solution

Let $S$ be defined on $\mathbb{R}^2$ by $S \left(x_{1}, x_{2}\right)=\left(x_{1} + x_{2}, x_{1} + 1 \right)$. Is $S$ a linear transformation? solution

Let $T\left(x_{1}, x_{2}\right)=\left(2x_{1} - x_{2}, x_{1}+ 3 x_{2},  5 x_{1} - 2 x_{2}\right)$. Show that $T$ is a one-to-one linear transformation. Does $T$ map $\mathbb{R}^{2}$ onto $\mathbb{R}^{3}$?   solution

Let $T$ be the linear transformation whose standard matrix is given. Decide if $T$ is a one-to-one mapping. Also decide if $T$ maps $\mathbb{R}^4$ onto $\mathbb{R}^4$.    solution

$\left[\begin{array}{cccc}{7} & {5} & {4} & {-9} \\ {10} & {6} & {16} & {-4} \\ {12} & {8} & {12} & {7} \\ {-8} & {-6} & {-2} & {5}\end{array}\right]$