Problem 2
Vectors \(\vec{x}\) and \(\vec{y}\) are given. Sketch \(\vec{x}, \vec{y}, \vec{x}+\vec{y},\) and \(\vec{x}-\vec{y}\) on the same Cartesian axes. $$ \vec{x}=\left[\begin{array}{l} 3 \\ 1 \end{array}\right], \vec{y}=\left[\begin{array}{c} 1 \\ -2 \end{array}\right] $$
Problem 2
A matrix \(A\) and vectors \(\vec{b}, \vec{u}\) and \(\vec{v}\) are given. Verify that \(\vec{u}\) and \(\vec{v}\) are both solutions to the equation \(A \vec{x}=\vec{b} ;\) that is, show that \(A \vec{u}=A \vec{v}=\vec{b}\). $$ \begin{array}{l} A=\left[\begin{array}{cc} 1 & -2 \\ -3 & 6 \end{array}\right] \\ \vec{b}=\left[\begin{array}{c} 2 \\ -6 \end{array}\right], \vec{u}=\left[\begin{array}{c} 0 \\ -1 \end{array}\right], \vec{v}=\left[\begin{array}{l} 2 \\ 0 \end{array}\right] \end{array} $$
Problem 3
Row and column vectors \(\vec{u}\) and \(\vec{v}\) are defined. Find the product \(\vec{u} \vec{v},\) where possible. $$ \vec{u}=\left[\begin{array}{ll} 1 & -1 \end{array}\right] \quad \vec{v}=\left[\begin{array}{l} 3 \\ 3 \end{array}\right] $$
Problem 3
Matrices \(A\) and \(B\) are given. Compute \((A B)^{-1}\) and \(B^{-1} A^{-1}\). $$ A=\left[\begin{array}{ll} 2 & 5 \\ 3 & 8 \end{array}\right], \quad B=\left[\begin{array}{cc} 1 & -1 \\ 1 & 4 \end{array}\right] $$
Problem 3
A matrix \(A\) and vectors \(\vec{b}, \vec{u}\) and \(\vec{v}\) are given. Verify that \(\vec{u}\) and \(\vec{v}\) are both solutions to the equation \(A \vec{x}=\vec{b} ;\) that is, show that \(A \vec{u}=A \vec{v}=\vec{b}\). $$ \begin{array}{l} A=\left[\begin{array}{ll} 1 & 0 \\ 2 & 0 \end{array}\right] \\ \vec{b}=\left[\begin{array}{l} 0 \\ 0 \end{array}\right], \vec{u}=\left[\begin{array}{c} 0 \\ -1 \end{array}\right], \vec{v}=\left[\begin{array}{c} 0 \\ 59 \end{array}\right] \end{array} $$
Problem 3
Vectors \(\vec{x}\) and \(\vec{y}\) are given. Sketch \(\vec{x}, \vec{y}, \vec{x}+\vec{y},\) and \(\vec{x}-\vec{y}\) on the same Cartesian axes. $$ \vec{x}=\left[\begin{array}{c} -1 \\ 1 \end{array}\right], \vec{y}=\left[\begin{array}{c} -2 \\ 2 \end{array}\right] $$
Problem 3
Matrices \(A\) and \(B\) are given below. Simplify the given expression. $$ A=\left[\begin{array}{cc} 1 & -1 \\ 7 & 4 \end{array}\right] \quad B=\left[\begin{array}{cc} -3 & 2 \\ 5 & 9 \end{array}\right] $$ $$ 3 A-A $$
Problem 3
Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ \begin{array}{l} A=\left[\begin{array}{ll} 3 & 3 \\ 6 & 4 \end{array}\right] \\ B=\left[\begin{array}{ll} 15 & -39 \\ 16 & -66 \end{array}\right] \end{array} $$
Problem 4
A matrix \(A\) and vectors \(\vec{b}, \vec{u}\) and \(\vec{v}\) are given. Verify that \(\vec{u}\) and \(\vec{v}\) are both solutions to the equation \(A \vec{x}=\vec{b} ;\) that is, show that \(A \vec{u}=A \vec{v}=\vec{b}\). $$ \begin{array}{l} A=\left[\begin{array}{ll} 1 & 0 \\ 2 & 0 \end{array}\right] \\ \vec{b}=\left[\begin{array}{l} -3 \\ -6 \end{array}\right], \vec{u}=\left[\begin{array}{l} -3 \\ -1 \end{array}\right], \vec{v}=\left[\begin{array}{l} -3 \\ 59 \end{array}\right] \end{array} $$
Problem 4
Vectors \(\vec{x}\) and \(\vec{y}\) are given. Sketch \(\vec{x}, \vec{y}, \vec{x}+\vec{y},\) and \(\vec{x}-\vec{y}\) on the same Cartesian axes. $$ \vec{x}=\left[\begin{array}{l} 2 \\ 0 \end{array}\right], \vec{y}=\left[\begin{array}{l} 1 \\ 3 \end{array}\right] $$
