Question

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

$\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}\mathbf{1}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{1}& \mathbf{1}\end{array}\mathbf{\right]}$

Short answer

Thus, the image of $T$is spanned by the vector of $\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$which geometrically represents a line in ${\mathrm{ℝ}}^3$.

Step-by-step solution

Obtain image of the transformation of the given matrix.

The given matrix is:

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$

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

Geometrical image of transformation.

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

localid="1664184139110" $$\begin{gathered} \mathit{T}\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\mathbf{=}\mathit{A}\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right] \\ =\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right] \\ =x_1\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]+x_2\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]+x_3\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right] \\ =\left(x_1+x_2+x_3\right)\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right] \end{gathered}$$

Hence, the image of$T$is spanned by the vector of$\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$which is geometrically represents a line in ${\mathrm{ℝ}}^3$.