Question

For each matrix in Exercise 17 through 22, describe the image of the transformation $\mathit{T}\mathbf{\left(}\overrightarrow{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathit{A}\overrightarrow{\mathbf{x}}$geometrically (as a line, plane, etc. in ${\mathbf{ℝ}}^{\mathbf{2}}\mathbf{}\mathbf{\text{and}}\mathbf{}{\mathbf{ℝ}}^{\mathbf{3}}$.

role="math" localid="1664181795984" $\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}\mathbf{2}& \mathbf{1}& \mathbf{3}\\ \mathbf{3}& \mathbf{4}& \mathbf{2}\\ \mathbf{6}& \mathbf{5}& \mathbf{7}\end{array}\mathbf{\right]}$

Short answer

Thus, the image of$T$consists of all linear combinations of the column vectors of$A$are$\overrightarrow{v_1}=\left[\begin{array}{c}2\\ 3\\ 6\end{array}\right],\overrightarrow{v_2}=\left[\begin{array}{c}1\\ 4\\ 5\end{array}\right],\overrightarrow{v_3}=\left[\begin{array}{c}3\\ 2\\ 7\end{array}\right]$.

Step-by-step solution

Obtain image of the transformation of the given matrix.

The given matrix is:

$A=\left[\begin{array}{ccc}2& 1& 3\\ 3& 4& 2\\ 6& 5& 7\end{array}\right]$

The objective is to describe the image of the transformation $T\left(\overrightarrow{x}\right)=A\overrightarrow{x}$geometrically.

The image of $A=\left[\begin{array}{ccc}2& 1& 3\\ 3& 4& 2\\ 6& 5& 7\end{array}\right]$consists of all vectors of the form:

$$\begin{gathered} \mathbf{T}\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\mathbf{=}\mathbf{A}\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right] \\ =\left[\begin{array}{ccc}2& 1& 3\\ 3& 4& 2\\ 6& 5& 7\end{array}\right]\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\\ {\mathrm{x}}_3\end{array}\right] \\ ={\mathrm{x}}_1\left[\begin{array}{c}2\\ 3\\ 6\end{array}\right]+{\mathrm{x}}_2\left[\begin{array}{c}1\\ 4\\ 5\end{array}\right]+{\mathrm{x}}_3\left[\begin{array}{c}3\\ 2\\ 7\end{array}\right] \end{gathered}$$

Thus, the vectorslocalid="1664181521724" $\left[\begin{array}{c}2\\ 3\\ 6\end{array}\right],\left[\begin{array}{c}1\\ 4\\ 5\end{array}\right],\left[\begin{array}{c}3\\ 2\\ 7\end{array}\right]$are the image of $T$.

The image of $T$consists of all linear combinations of the column vectors of $A$.

$\overrightarrow{v_1}=\left[\begin{array}{c}2\\ 3\\ 6\end{array}\right],\overrightarrow{v_2}=\left[\begin{array}{c}1\\ 4\\ 5\end{array}\right],\overrightarrow{v_3}=\left[\begin{array}{c}3\\ 2\\ 7\end{array}\right]$

Hence, the image of $T$consists of all linear combinations of the column vectors of $A$arelocalid="1664181836079" $\overrightarrow{v_1}=\left[\begin{array}{c}2\\ 3\\ 6\end{array}\right],\overrightarrow{v_2}=\left[\begin{array}{c}1\\ 4\\ 5\end{array}\right],\overrightarrow{v_3}=\left[\begin{array}{c}3\\ 2\\ 7\end{array}\right]$.

Graph of image of transformation.

Thus, the image of $T$is the plane$V$ spanned by the three vectors $\overrightarrow{v_1}$, $\overrightarrow{v_2}$and $\overrightarrow{v_3}$.

That is, the plane through the origin and the end points $\left(2,3,6\right),\left(1,4,5\right),\left(3,2,7\right)$of $\overrightarrow{v_1},\overrightarrow{v_2}$,and$\overrightarrow{v_3}$ respectively

Thus, the geometrically the transformation represents all ${\mathrm{ℝ}}^3$.