(Solved): 1. Determine the domain and the codomain (range) of the transformation in each of the following rep ...

1. Determine the domain and the codomain (range) of the transformation in each of the following representations. (a) $\begin{array}{l}{w}_1=2x_1+7x_2-4x_3\\ w_2=4x_1-3x_2+2x_3\end{array}$ (b) $\left[\begin{array}{rrr}2& 1& -6\\ 3& 7& -4\\ 1& 0& 3\end{array}\right]\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]$ (c) $T\left(x_1,x_2,x_3\right)=\left(4x_1+x_2,x_1+x_2\right)$ 2. Write out the standard matrix for the transformation defined by the following equations: $\begin{array}{l}{w}_1=7x_1+2x_2-8x_3\\ w_1=x_1\\ w_2=-x_2+5x_3\\ w_2=x_1+x_2\\ w_3=4x_1+7x_2-x_3\\ w_3=x_1+x_2+x_3\\ w_4=x_1+x_2+x_3+x_4\end{array}$ (b) 3. Write out the standard matrix for the transformation $T$ defined by the following formulas: (a) $T\left(x_1,x_2,x_3,x_4\right)=\left(7x_1+2x_2-x_3+x_4,x_2+x_3,-x_1\right)$ (b) $T\left(x_1,x_2,x_3,x_4\right)=\left(x_4,x_1,x_3,x_2,x_1-x_3\right)$ (c) $T\left(x_1,x_2,x_3\right)=\left(x_1+2x_2+x_3,x_1+5x_2,x_3\right)$ (d) $T\left(x_1,x_2,x_3\right)=\left(4x_1,7x_2,-8x_3\right)$ 4. The images of standard basis vectors for ${\mathbb{R}}^3$ are given for the linear transformation $T:{\mathbb{R}}^3\to$ ${\mathbb{R}}^3$. Find the standard matrix for the transformation, and find $T(\vec{x})$. (a) $T\left(\overrightarrow{e_1}\right)=\left[\begin{array}{l}1\\ 3\\ 0\end{array}\right],T\left(\overrightarrow{e_2}\right)=\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right],T\left(\overrightarrow{e_3}\right)=\left[\begin{array}{r}4\\ -3\\ -1\end{array}\right];\vec{x}=\left[\begin{array}{l}2\\ 1\\ 0\end{array}\right]$ (b) $T\left(\overrightarrow{e_1}\right)=\left[\begin{array}{l}2\\ 1\\ 3\end{array}\right],T\left(\overrightarrow{c_2}\right)=\left[\begin{array}{r}-3\\ -1\\ 0\end{array}\right],T\left(\overrightarrow{e_3}\right)=\left[\begin{array}{l}1\\ 0\\ 2\end{array}\right];\vec{x}=\left[\begin{array}{l}3\\ 2\\ 1\end{array}\right]$ 5. Let $L(x,y,z)=(x+z;y,x-z)$. Show that $L$ is a linear transformation. 6. Let $L\left(\left[\begin{array}{l}{u}_1\\ u_2\\ u_3\end{array}\right]\right)=\left[\begin{array}{c}{u}_1+1\\ u_2-u_3\end{array}\right]$. Show that $L$ fails to be a linear transformation.