Linear Transformation

  1. 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

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

  2. 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

  3. 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

  4. 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

  5. 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

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