Given vector b, is vector x in the range of the linear transformation

Question

I'm given a matrix $$A=\begin{pmatrix} 1&{-4}&7&{-5}\\0&1&{-4}&3\\2&{-6}&6& {-4}\end{pmatrix}$$

I'm asked to find all vectors $x$ in $R^4$ that are mapped onto the zero vector by the transformation $x \to Ax$.

I found the answer to be $$\begin{pmatrix}x_1\\x_2\\x_3\\x_4\\\end{pmatrix}=c_1\begin{pmatrix}9\\4\\1\\0\end{pmatrix}+c_2\begin{pmatrix}{-7}\\{-3}\\0\\1\\\end{pmatrix}$$

The next question, which I am having trouble with, is

Further, given $b =\begin{matrix} [-1 & 1 & 0]\end{matrix} $, is $x$ in the range of the linear transformation $x \to Ax$?

How do I solve this?

Answer 1

The standard way for the second question is to consider the augmented matrix , with vector $b$ as last column, and perform the Gauss algorithm on it in order to obtain the RREF.

If you obtain a consistent system of equivalent equations then $b$ is a special solution of $Ax=b$.