Checking Linear Dependency

Question

Suppose the non zero $n \times 1$ column vector $x$ solves the system of equations $Ax=b$ where $A$ is $m \times n$ matrix whose columns are the vectors $a_1,a_2,\ldots,a_n$ and $b$ is a $m \times 1$ column vector. then the set of the vectors $\{a_1, a_2, \ldots, a_n, b\}$ is Linearly dependent . Please explain how? Shouldn't there be conditon the set of vectors in "$A$" matrix ?

Answer 1

Note that $$Ax = b \iff \begin{bmatrix} a_1 & ... & a_n\end{bmatrix}\begin{bmatrix}x_1\\ \vdots \\ x_n\end{bmatrix} - b = 0 \iff \sum_{i=1}^n x_i a_i - b = \sum_{i=1}^n x_i a_i + (-1)\times b =0.$$

Since $-1\ne 0$, this proves that $(a_1,...,a_n,b)$ are linearly dependent.