How unique are solutions of a linear system of equations?

Let’s look at the following linear systems of equations in three variables $x_1,x_2,x_3\in\R$ :

System 1: Consider

\displaystyle \def\arraystretch{1.25} \begin{aligned} 3x_1+x_2&=5\\ x_1+4x_2+x_3&=-1\\ x_1&=1 \end{aligned}

It has the unique solution $x_{1} = 1, x_{2} = 2, x_{3} = - 10$ .

System 2: Consider

\displaystyle \def\arraystretch{1.25} \begin{aligned} 3x_1+x_2&=5\\ x_1+4x_2+x_3&=-1\\ 6x_1+2x_2&=10 \end{aligned}

For each $x_1 \in \R$ it has the solution $x_{2} = 5 - 3 x_{1}, x_{3} = - 21 + 11 x_{1}$ . Hence there are infinitely many solutions.

System 3: Consider

\displaystyle \def\arraystretch{1.25} \begin{aligned} 3x_1+x_2&=5\\ 6x_1+2x_2&=10\\ -3x_1-x_2&=-5 \end{aligned}

Here we can choose $x_{2} = 5 - 3 x_{1}$ and get a solution for arbitrary $x_1, x_3\in\R$ .

If there are more redundant rows then the linear system of equations has more solutions.

System 1: unique solution; represents a point in $R^{3}$
System 2: solution for any $x_{1}$ ; solutions represent a straight line in $R^{3}$
System 3: solution for any choice of $x_{1}$ and $x_{3}$ ; represents a plane in $R^{3}$

If we have more than one solution, what do they have in common?

Let $x ’$ and $x ’’$ be two solutions of a linear system of equations $A x = y$ . What is the difference between the two solutions $x ’$ and $x ’’$ ? We apply $A$ to the difference $x ’ - x ’’$ : $A (x ’ - x ’’) = A x ’ - A x ’’ = y - y = 0$ .

The more solutions to the system $A x = y$ there are, the more vectors we can find that get mapped to zero. Hence the “amount” of x with $A x = 0$ is a measure for the uniqueness of the solution $A x = y$ for any $y$ . For that reason, we give it a name: The kernel of $A$ . We write it as

\displaystyle \ker(A) = \{x \in \R^n \mid Ax = 0\}.

The bigger the kernel the more solutions exist. If $\ker(A)=0$ the solution is unique.

5How unique are solutions of a linear system of equations?