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If A is an$n脳m$matrix with row vectors $\stackrel{鈫�}{蝇1},...,\stackrel{鈫�}{蝇n}$and $\stackrel{鈫�}{x}$is a vector in ${\mathrm{鈩�}}^m$then,

$\stackrel{鈫�}{Ax}=\left[\begin{array}{c}鈭�{\stackrel{鈫�}{蝇}}_1鈭�\\ 鈰�\\ {\stackrel{鈫�}{蝇}}_n鈭�\end{array}\right]\stackrel{鈫�}{x}=\left[\begin{array}{c}鈭�{\stackrel{鈫�}{蝇}}_1鈰�\stackrel{鈫�}{x}鈭�\\ 鈰�\\ 鈭�{\stackrel{鈫�}{蝇}}_n鈰�\stackrel{鈫�}{x}鈭�\end{array}\right]$

Consider a linear system,

$A\stackrel{鈫�}{x}=\stackrel{鈫�}{b}$

When $\stackrel{鈫�}{b}=0,A\stackrel{鈫�}{x}=0$

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

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.

Given:

$A\stackrel{鈫�}{x}=0$ is a homogeneous linear system.

$\stackrel{鈫�}{x_1}$and$\stackrel{鈫�}{x_2}$are homogeneous solutions.

Consider the system,

$\begin{array}{r}A\stackrel{鈫�}{x}=0\\ 鈬�A\left({\stackrel{鈫�}{x}}_1+\stackrel{鈫�}{x_2}\right)=0\\ 鈬�A{\stackrel{鈫�}{x}}_1+A{\stackrel{鈫�}{x}}_2=0\\ 鈭�{\stackrel{鈫�}{x}}_1+{\stackrel{鈫�}{x}}_2=0\end{array}$

Given:

$A\stackrel{鈫�}{x}=0$is a homogeneous linear system.

$\stackrel{鈫�}{x}$ is a homogeneous solution.

k is a arbitrary constant.

Consider the system,

$\begin{array}{r}A\stackrel{鈫�}{x}=0\\ 鈬�A\left(k\stackrel{鈫�}{x}\right)=0\\ 鈬�k\left(\stackrel{鈫�}{Ax}\right)=0\\ 鈬�k\left(0\right)\stackrel{鈫�}{x}=0\\ 鈭�\stackrel{鈫�}{kx}=0\end{array}$

a. $\stackrel{鈫�}{x}=0$is a solution.

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

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

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