Question

Question:A linear system of the form$\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{=}\mathbf{0}$ is called homogeneous. Justify the following facts:

a.All homogeneous systems are consistent.

b.A homogeneous system with fewer equations than unknowns has infinitely many solutions.

c.If$\overrightarrow{{\mathbf{x}}_{\mathbf{1}}}\mathbf{}\mathbf{and}\mathbf{}\overrightarrow{{\mathbf{x}}_{\mathbf{2}}}$ are solutions of the homogeneous system$\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{=}\mathbf{0}$, then$\overrightarrow{{\mathbf{x}}_{\mathbf{1}}}\mathbf{+}\overrightarrow{{\mathbf{x}}_{\mathbf{2}}}$ is a solution as well.

d.If$\overrightarrow{\mathbf{x}}$ is a solution of the homogeneous system$\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{=}\mathbf{0}$ andkis an arbitrary constant, then$k\overrightarrow{x}$ is a solution as well.

Short answer

a.$\overrightarrow{x}=0$ is a solution for a linear system of the form and hence all the homogeneous systems are consistent.

b. Using theorem 1.3.3., homogeneous system with lesser equations than unknowns have infinitely many solutions.

c. Yes, $\overrightarrow{{\mathrm{x}}_1}+\overrightarrow{{\mathrm{x}}_2}$is also a solution, if $\overrightarrow{{\mathrm{x}}_1}and\overrightarrow{{\mathrm{x}}_2}$are solutions of the homogeneous system $A\overrightarrow{x}=0$.

d. Yes, $k\overrightarrow{x}$is also a solution, if $\overrightarrow{x}$is a solution of the homogeneous system $A\overrightarrow{x}=0$.

Step-by-step solution

(a) Consider the system.

If A is an$n\times m$matrix with row vectors $\overrightarrow{\omega 1},...,\overrightarrow{\omega n}$and $\overrightarrow{x}$is a vector in ${\mathrm{ℝ}}^m$then,

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

Consider a linear system,

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

Determine the solution for the system.

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

Thus, $\overrightarrow{x}=0$will also be a solution for homogeneous system, $A\overrightarrow{x}=0$.

(b) Consider a theorem

Theorem 1.3.3

a. For a linear system with at least one solution, then, the number of unknowns must be equal to the number of equations.

b. A linear system will have no solution or infinitely many solutions for number of equations less than the number of unknowns.

Considering the theorem 1.3.3, it can be stated that, a homogeneous system with fewer equations than unknowns has infinitely many solutions.

(c) Consider the system.

Given:

$A\overrightarrow{x}=0$ is a homogeneous linear system.

$\overrightarrow{x_1}$and$\overrightarrow{x_2}$are homogeneous solutions.

Consider the system,

$\begin{array}{r}A\overrightarrow{x}=0\\ \Rightarrow A\left({\overrightarrow{x}}_1+\overrightarrow{x_2}\right)=0\\ \Rightarrow A{\overrightarrow{x}}_1+A{\overrightarrow{x}}_2=0\\ \therefore {\overrightarrow{x}}_1+{\overrightarrow{x}}_2=0\end{array}$

Step 5:(d) Consider the system.

Given:

$A\overrightarrow{x}=0$is a homogeneous linear system.

$\overrightarrow{x}$ is a homogeneous solution.

k is a arbitrary constant.

Consider the system,

$\begin{array}{r}A\overrightarrow{x}=0\\ \Rightarrow A\left(k\overrightarrow{x}\right)=0\\ \Rightarrow k\left(\overrightarrow{Ax}\right)=0\\ \Rightarrow k\left(0\right)\overrightarrow{x}=0\\ \therefore \overrightarrow{kx}=0\end{array}$

Final answer

a. $\overrightarrow{x}=0$is a solution.

b. Using theorem 1.3.3., homogeneous system with lesser equations than unknowns have infinitely many solutions.

c. Yes, $\overrightarrow{x_1}+\overrightarrow{x_2}$is also a solution, if $\overrightarrow{{\mathrm{x}}_1}\mathrm{and}\overrightarrow{x_2}$are solutions of the homogeneous system $A\overrightarrow{x}=0$.

d. Yes, $k\overrightarrow{x}$ is also a solution, if $\overrightarrow{x}$is a solution of the homogeneous system $A\overrightarrow{x}=0$.