Consistency of Linear System by its Constant Vector

Theorem

A linear system $A\mathbf{x}=\mathbf{b}$ is consistent if and only if $\mathbf{b}$ is in column space of $A$

Example

Let $A\mathbf{x}=\mathbf{b}$ be the linear system

$$\left[\begin{array}{ccc}-1& 3& 2\\ 1& 2& -3\\ 2& 1& -2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}1\\ -9\\ -3\end{array}\right]$$

Show that $\mathbf{b}$ is in the column space of $A$ by expressing it as a linear combination of the column vectors of $A$.

Solving the system by Gaussian elimination yields (verify)

$$x_1=2,x_2=-1,x_3=3$$

It follows from this that

$$2\left[\begin{array}{c}-1\\ 1\\ 2\end{array}\right]-\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right]+3\left[\begin{array}{c}2\\ -3\\ -2\end{array}\right]=\left[\begin{array}{c}1\\ -9\\ -3\end{array}\right]$$

$\mathbf{b}=(2,-1,3)$ is a solution to $A\mathbf{x}=\mathbf{b}$. Therefore, $\mathbf{b}$ is in column space of $A$.