Question

Express the image of the matrix

$\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cccc}\mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{6}\\ \mathbf{1}& \mathbf{2}& \mathbf{3}& \mathbf{4}\\ \mathbf{1}& \mathbf{3}& \mathbf{5}& \mathbf{2}\\ \mathbf{1}& \mathbf{4}& \mathbf{7}& \mathbf{0}\end{array}\mathbf{\right]}$

as the kernel of a matrix$\mathit{B}$. Hint: The image of$\mathit{A}$consists of all vectors$\overrightarrow{\mathbf{y}}$in${\mathbf{ℝ}}^{\mathbf{4}}$such that the system$\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{=}\overrightarrow{\mathbf{y}}$is consistent. Write this system more explicitly:

localid="1664197199135" $\left|\begin{array}{l}{x}_1+ x_2+ x_3+6x_4=y_1\\ x_1+2x_2+3x_3+4x_4= y_2\\ x_1+3x_2+5x_3+2x_4= y_3\\ x_1+4x_2+7x_{3 } = y_4\end{array}\right|$.

Now, reduce rows:

$\left|\begin{array}{rcl}{\mathit{x}}_{\mathbf{1}}\mathbf{-}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }{\mathit{x}}_{\mathbf{3}}\mathbf{+}\mathbf{8}{\mathit{x}}_{\mathbf{4}}& \mathbf{=}& \mathbf{}\mathbf{}\mathbf{}\mathbf{4}{\mathit{y}}_{\mathbf{3}}\mathbf{-}\mathbf{3}{\mathit{y}}_{\mathbf{4}}\\ {\mathit{x}}_{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{x}}_{\mathbf{3}}\mathbf{-}\mathbf{2}{\mathit{x}}_{\mathbf{4}}\mathbf{}\mathbf{}\mathbf{}& \mathbf{=}& \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}{\mathit{y}}_{\mathbf{3}}\mathbf{+}{\mathit{y}}_{\mathbf{4}}\\ \mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}& \mathbf{=}& \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{3}{\mathit{y}}_{\mathbf{3}}\mathbf{+}\mathbf{2}{\mathit{y}}_{\mathbf{4}}\\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{}\mathbf{ }\mathbf{ }{\mathit{y}}_{\mathbf{2}}\mathbf{-}\mathbf{ }\mathbf{2}{\mathit{y}}_{\mathbf{3}}\mathbf{+}{\mathit{y}}_{\mathbf{4}}\end{array}\right|$

For which the vectors $\overrightarrow{\mathbf{y}}$is this system consistent? The answer allows you to express im ( $\mathit{A}$) as the kernel of a$\mathbf{2}\mathbf{\times }\mathbf{4}$matrix $\mathit{B}$.

Short answer

The $\mathrm{im}\left(\mathrm{A}\right)=\ker \left(\mathrm{B}\right)$, here the matrix $\mathrm{B}$is, $B=\left[\begin{array}{cccc}1& 0& -3& 2\\ 0& 1& -2& 1\end{array}\right]$.

Step-by-step solution

Define kernel of linear transformation

Thekernel of linear transformation is defined as follows:

The kernel of a linear transformation$T\left(\overrightarrow{x}\right)=A\overrightarrow{x}$from${\mathrm{ℝ}}^m$to${\mathrm{ℝ}}^n$consists of all zeros of the transformation, i.e., the solutions of the equations $T\left(\overrightarrow{x}\right)=A\overrightarrow{x}=0$.

It is denoted by ker$\left(T\right)$or ker$\left(A\right)$.

Let$A$be the matrix given by,

$A=\left[\begin{array}{cccc}1& 1& 1& 6\\ 1& 2& 3& 4\\ 1& 3& 5& 2\\ 1& 4& 7& 0\end{array}\right]$.

 Express the image of the matrix A as the kernel of a matrix B

From the hint given in the question, we can say that $\overrightarrow{y}\in im\left(A\right)$if,

$$\begin{gathered} \left[\begin{array}{c}{y}_1-3y_2+2y_4\\ y_2-2y_3+y_4\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right] \\ \begin{array}{c}{y}_1-3y_3+2y_4=0\\ y_2-2y_3+y_4=0\end{array} \end{gathered}$$.

Thus,$\mathrm{im}\left(\mathrm{A}\right)=\ker \left(\mathrm{B}\right)$, where the matrix$B$is,

localid="1664198103031" $B=\left[\begin{array}{cccc}1& 0& -3& 2\\ 0& 1& -2& 1\end{array}\right]$.