So far we have seen 2 ways of seeing $Ax = b$

Picture 1 (The Row Picture)

Think this like solution to a system of linear equation

The Matrix

Graphical Representation

Picture 2 (The Column Picture)

Think this like linear combination of vectors

The Matrix

Graphical Representation

The New Picture (Transformation)

Think the matrix $A$ as an operator for Transformation . $A$ “acts” on a vector $x$ by multiplication to produce a new vector called $Ax$ .

An example of Sheer Transformation

Example