Question

Consider a solution$\overrightarrow{{\mathbf{x}}_{\mathbf{1}}}$of the linear system$\mathit{A}\overrightarrow{\mathbf{x}}=\overrightarrow{b}$. Justify the facts stated in parts (a) and (b):

a. If${\overrightarrow{\mathbf{x}}}_{\mathbf{h}}$is a solution of the system$\mathit{A}\overrightarrow{\mathbf{x}}=\overrightarrow{0}$, then$\overrightarrow{{\mathbf{x}}_{\mathbf{1}}}\mathbf{+}\overrightarrow{{\mathbf{x}}_{\mathbf{h}}}$ is a solution of the system$\mathbf{A}\mathbf{=}\overrightarrow{\mathbf{x}}\mathbf{=}\overrightarrow{\mathbf{b}}$.

b. If$\overrightarrow{{\mathbf{x}}_{\mathbf{2}}}$is another solution of the system$\mathit{A}\overrightarrow{\mathbf{x}}=\overrightarrow{b}$, then$\overrightarrow{{\mathbf{x}}_{\mathbf{1}}}\mathbf{+}\overrightarrow{{\mathbf{x}}_{\mathbf{h}}}$is a solution of the system $\mathbf{A}\overrightarrow{\mathbf{x}}\mathbf{+}\overrightarrow{0}$.

c. Now suppose A is a$\mathbf{2}\mathbf{\times }\mathbf{2}$matrix. A solution vector$\overrightarrow{{\mathbf{x}}_{\mathbf{1}}}$of the system$\mathbf{A}\overrightarrow{\mathbf{x}}\mathbf{+}\overrightarrow{b}$is shown in the accompanying figure. We are told that the solutions of the system$\mathbf{A}\overrightarrow{\mathbf{x}}\mathbf{=}\overrightarrow{\mathbf{0}}$form the line shown in the sketch. Draw the line consisting of all solutions of the system$\mathbf{A}\overrightarrow{\mathbf{x}}\mathbf{=}\overrightarrow{b}$.

If you are puzzled by the generality of this problem, think about an example first:

$\mathbf{A}\mathbf{=}\left(\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}6\end{array}\right)\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{=}\left[\begin{array}{l}3\\ 9\end{array}\right]\mathbf{and}\overrightarrow{{\mathbf{x}}_{\mathbf{1}}}\mathbf{=}\left[\begin{array}{l}1\\ 1\end{array}\right]$

Short answer

a.$\overrightarrow{{\mathrm{x}}_1}+\overrightarrow{{\mathrm{x}}_{\mathrm{h}}}$is a solution of the system$A\overrightarrow{\mathrm{x}}=\overrightarrow{b}$.

b.$\overrightarrow{{\mathrm{x}}_2}+\overrightarrow{{\mathrm{x}}_1}$is a solution of the system$A\overrightarrow{\mathrm{x}}=\overrightarrow{0}$.

c. The line consisting of all the solutions of the system$A\overrightarrow{\mathrm{x}}=\overrightarrow{b}$is,

Step-by-step solution

Consider the system.

If is an matrix with row vectors ${\overrightarrow{\omega }}_1,.....,{\overrightarrow{\omega }}_n$and $\overrightarrow{x}$is a vector in ${\mathrm{ℝ}}^m$then,

$A\overrightarrow{x}=\left[\begin{array}{c}-{\overrightarrow{\omega }}_1-\\ ⋮\\ {\overrightarrow{\omega }}_n-\end{array}\right]\overrightarrow{x}=\left[\begin{array}{c}-{\overrightarrow{\omega }}_1\cdot \overrightarrow{x}-\\ ⋮\\ -{\overrightarrow{\omega }}_n\cdot \overrightarrow{x}-\end{array}\right]$

Consider a linear system,

$A\overrightarrow{x}=\overrightarrow{b}$

Determine the solution for the system.

${\overrightarrow{x}}_1$is a solution of the system, $A\overrightarrow{x}=\overrightarrow{b}$.......(1).

${\overrightarrow{x}}_h$is a solution of the system, $A\overrightarrow{x}=\overrightarrow{0}$.

When $\overrightarrow{b}=0,A\overrightarrow{x}=\overrightarrow{b}$,

Thus, $\overrightarrow{x_h}$is a solution of the system, $A\overrightarrow{x}=\overrightarrow{b}.....\left(2\right)$

From the equations (1) and (2), it’s clear that,

$\overrightarrow{x_1}+\overrightarrow{x_h}$is also a solution of the system $A\overrightarrow{x_1}+\overrightarrow{b}$.

Determine the solution for the system.

${\overrightarrow{x}}_1$is a solution of the system, $A\overrightarrow{x}=\overrightarrow{b}$.

${\overrightarrow{x}}_2$is a solution of the system, $A\overrightarrow{x}=\overrightarrow{b}$.

When $\overrightarrow{b}=0$,

$A\overrightarrow{x}=\overrightarrow{b}\Rightarrow A\overrightarrow{x}\to =\overrightarrow{0}$

Thus, $\overrightarrow{x_1}$is a solution of the system,$A\overrightarrow{x_1}=\overrightarrow{0}$…… (1)

is a solution of the system,$A\overrightarrow{x}=\overrightarrow{0}$……(2)

From the equations (1) and (2), it’s clear that,

$\overrightarrow{x_2}={\overrightarrow{x}}_1$is also a solution of the system $A\overrightarrow{x}=\overrightarrow{0}$.

Consider the system

The line representing the system$A\overrightarrow{x}=\overrightarrow{0}$must be parallel to the line representing the system$A\overrightarrow{x}=\overrightarrow{b}$and must pass through the solution${\overrightarrow{x}}_1$.

Thus, the line representing the system$A\overrightarrow{x}=\overrightarrow{b}$is,