5How 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

It has the unique solution $x_1=1, x_2=2, x_3 = -10$.

System 2: Consider

For each $x_1 \in \R$ it has the solution $x_2 = 5-3x_1, x_3 = -21+11x_1$. Hence there are infinitely many solutions.

System 3: Consider

Here we can choose $x_2=5-3x_1$ 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 $Ax=y$. What is the difference between the two solutions $x’$ and $x’’$? We apply $A$ to the difference $x’-x’’$: $A(x’-x’’) = Ax’ - Ax’’ = y-y = 0$.

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

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