Introduction: It was immensely frustrating to have to retake linear algebra multiple times in my life. It was even more frustrating to have forgotten computation rules for gaussian elimination, and how one uses them to find properties of the linear transformation such as rank, null space or solution space. The issue is as follows: Gaussian elimination is almost always taught as the first subject of any linear algebra course, followed by the interpretation of a matrix as a linear transformation. After such an interpretation is given and the isomorphism defined, courses rarely return to the gaussian elimination algorithm, to explain what happens to the linear transformation when gaussian elimination is performed. I hope to bridge this gap in this text.