Notes from Prof. Gilbert Strang’s Lecture on MIT OpenCourseWare: The Geometry of Linear Equations

In linear algebra, when faced with equations, we often try to visualize them using the row picture.

Row Picture: In 2-D, we can think of it as a line and aim to find its intersection.

Column Picture: We aim to find the weights of a linear combination of columns. In the image on the right, we see the addition of two vectors. We start from the origin and add them at the tail.

As we move to higher dimensions, the row picture becomes more challenging to visualize, while the column picture remains straightforward.

Furthermore, the column picture allows us to check if the combination of all columns fills the entire space. We can verify this through elimination.

Additionally, we can develop a habit of viewing matrix multiplication as a linear combination of columns.