The number of solutions to a system of linear equations Can anyone suggest a formal proof that a system of linear equations can have no solution, one solution or infinitely many solutions? 
 A: Your system is $Ax=b$. It may have no solutions. It may have exactly one solution. But, if it has at least two solutions $x_1$ and $x_2$, we can define $x_t=tx_1+(1-t)x_2$. Then 
$$
Ax_t=tAx_1+(1-t)Ax_2=tb+(1-t)b=b,
$$
so $x_t$ is a solution for all $t\in\mathbb R$, and so the system has infinitely many solutions. 
A: Consider a linear system in $\mathrm x \in \mathbb R^n$
$$\mathrm A \mathrm x = \mathrm b$$
where $\mathrm A \in \mathbb R^{m \times n}$ and $\mathrm b \in \mathbb R^m$ are given. Suppose that the system is feasible and that $\mathrm x^{(1)}$ and $\mathrm x^{(2)}$ are two solutions. Hence, $\mathrm A \mathrm x^{(1)} = \mathrm b$ and $\mathrm A \mathrm x^{(2)} = \mathrm b$. Subtracting these two, we obtain
$$\mathrm A \mathrm x^{(1)} - \mathrm A \mathrm x^{(2)} = \mathrm b - \mathrm b = 0_m$$
or,
$$\mathrm A (\mathrm x^{(1)} - \mathrm x^{(2)}) = 0_m$$
Hence, $\mathrm x^{(1)} - \mathrm x^{(2)}$ is in the null space of $\mathrm A$. If


*

*the null space is trivial (i.e., it contains only $0_n$), then $\mathrm x^{(1)} - \mathrm x^{(2)} = 0_n$, or, $\mathrm x^{(1)} = \mathrm x^{(2)}$.

*the null space is nontrivial (i.e., it is not $0$-dimensional), then any affine combination of $\mathrm x^{(1)}$ and $\mathrm x^{(2)}$ is also a solution to the linear system, for the following holds for all $\gamma \in \mathbb R$
$$\mathrm A (\gamma \mathrm x^{(1)} + (1-\gamma) \mathrm x^{(2)}) = \gamma \mathrm A \mathrm x^{(1)} + (1-\gamma) \mathrm A \mathrm x^{(2)} = \gamma \mathrm b + (1-\gamma) \mathrm b = \mathrm b$$
Thus, if a linear system is feasible, it either has one solution, or it has infinitely many.
A: What you are asking for is an example. In $\mathbb{R}^1$, consider


*

*$0\cdot x=1$;

*$x=2016$;

*$x=x$.


In $\mathbb{R}^2$, consider  


*

*$0\cdot x+0\cdot y=1$;  

*$$\begin{cases}
x=2016\\
y=2016
\end{cases}
$$  

*$x=y$;  


Can you see that how this example could be generalized to $\mathbb{R}^n$?
