# Homogeneous Linear Systems

For a nonhomogeneous system of 2012 equations in 1999 unknowns, answer the following three questions:

• Can the system be inconsistent?
• Can the system have infinitely many solutions?
• Can the system have a unique solution?
• What have you tried? (And for that matter, what have you learned?) See meta.math.stackexchange.com/questions/1803/… for why I ask. – Willie Wong Nov 2 '12 at 16:00
• What do you know about linear systems? – Emily Nov 2 '12 at 16:01
• Honestly not much. I'm hoping to learn from answers – Sam Nov 2 '12 at 16:03
• This really is incredibly similar to your other question here math.stackexchange.com/questions/227608/… Would it not be better to ask one question and see if you get the idea? – Simon Hayward Nov 2 '12 at 16:12
• Simplify! Consider, say 6 equations and 3 variables. – vesszabo Nov 2 '12 at 16:32

There's homogeneous in the title and nonhomogeneous in the text. In either case, we have a $2012 \times 1999$ matrix $A$. We are looking for solutions to $A\mathbf{x}=\mathbf{0}$ (homogeneous) or $A\mathbf{x}=\mathbf{b}$ where $\mathbf{b} \neq \mathbf{0}$ (inhomogeneous).

## Homogeneous

We immediately have $A\mathbf{0}=\mathbf{0}$, so $\mathbf{x}=\mathbf{0}$ is a solution, so the system is consistent. (In fact, all homogeneous systems of linear equations are consistent due to the all-$0$ solution.)

By definition, it will have infinitely many solutions if and only if the null space of $A$ is non-trivial, i.e., if and only if the rank of $A$ is less than $1999$ (by the Rank-Nullity Theorem). One example is when $[A|\mathbf{0}]$ corresponds to the system of equations \begin{align*} x_1 &=0 \\ 2x_1 &=0 \\ 3x_1 &=0 \\ \vdots \\ 2012x_1 &=0. \end{align*}

By definition, it will have a unique solution (and this solution must be the zero solution) if and only if the null space of $A$ is trivial, i.e., if and only if the rank of $A$ is equal to $1999$. One example is when $A$ has $1$'s on the main diagonal and $0$'s elsewhere, which has rank $1999$. This is when $[A|\mathbf{0}]$ corresponds to the system of equations \begin{align*} x_1 &=0 \\ x_2 &=0 \\ \vdots \\ x_{1999} &=0 \\ 0 &=0 \\ \vdots \\ 0 &=0. \\ \end{align*}

## Inhomogeneous

It can be inconsistent: e.g. when $[A|\mathbf{b}]$ corresponds to the system of equations \begin{align*} x_1+x_2+\cdots+x_{1999} &=1 \\ x_1+x_2+\cdots+x_{1999} &=2 \\ \vdots \\ x_1+x_2+\cdots+x_{1999} &=2012. \\ \end{align*}

It can have infinitely many solutions: e.g. when $[A|\mathbf{b}]$ corresponds to the system of equations \begin{align*} x_1 &=1 \\ 2x_1 &= 2 \\ 3x_1 &= 3 \\ \vdots \\ 2012x_1 &= 2012. \\ \end{align*}

It can have a unique solution: e.g. when $[A|\mathbf{b}]$ corresponds to the system of equations \begin{align*} x_1 &=1 \\ x_2 &=1 \\ \vdots \\ x_{1999} &=1 \\ 0 &=0 \\ \vdots \\ 0 &=0. \\ \end{align*}

Note: The answers to the second part changes if we work over finite fields.