Question

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

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}4& 7& 3\\ 1& 9& 2\\ 5& 6& 8\end{array}\right]$

Short answer

Thus, the image of $T$consists of all linear combinations of the column vectors of$A$are$\overrightarrow{v_1}=\left[\begin{array}{c}4\\ 1\\ 5\end{array}\right],\overrightarrow{v_2}=\left[\begin{array}{c}7\\ 9\\ 6\end{array}\right],\overrightarrow{v_3}=\left[\begin{array}{c}3\\ 2\\ 8\end{array}\right]$.

Step-by-step solution

Obtain image of the transformation of the given matrix.

The given matrix is:

$A=\left[\begin{array}{ccc}4& 7& 3\\ 1& 9& 2\\ 5& 6& 8\end{array}\right]$

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

The image of localid="1664183235646" $A=\left[\begin{array}{ccc}4& 7& 3\\ 1& 9& 2\\ 5& 6& 8\end{array}\right]$consists of all vectors of the form:

Thus, the vectors localid="1664183456532" $\left[\begin{array}{c}4\\ 1\\ 5\end{array}\right],\left[\begin{array}{c}7\\ 6\\ 9\end{array}\right],\left[\begin{array}{c}3\\ 2\\ 8\end{array}\right]$are the image of $T$.

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

localid="1664183551134" $\overrightarrow{v_1}=\left[\begin{array}{c}4\\ 1\\ 5\end{array}\right],\overrightarrow{v_2}=\left[\begin{array}{c}7\\ 9\\ 6\end{array}\right],\overrightarrow{v_3}=\left[\begin{array}{c}3\\ 2\\ 8\end{array}\right]$

Hence, the image of$T$consists of all linear combinations of the column vectors of$A$are localid="1664183625644" $\overrightarrow{v_1}=\left[\begin{array}{c}4\\ 1\\ 5\end{array}\right],\overrightarrow{v_2}=\left[\begin{array}{c}7\\ 9\\ 6\end{array}\right],\overrightarrow{v_3}=\left[\begin{array}{c}3\\ 2\\ 8\end{array}\right]$.

Geometrical 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$\left(4,1,5\right),\left(7,9,6\right),\left(3,2,8\right)$of $\overrightarrow{v_1}$, $\overrightarrow{v_2}$and$\overrightarrow{v_3}$ respectively

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