We more or less conclude the formalization of solving systems of linear equations by discussing the actual solutions to such a system. Two types of systems are distinguished: the homogeneous systems of the form Ax = 0, and the non-homogeneous systems Ax = b with non-zero constant term. We characterize the solutions of the non-homogeneous systems by writing them in terms of the solutions of the homogeneous systems, and thus it suffices to be able to solve the homogeneous system. (Note the analogy with solutions of differential equations!) Finally, we define the very important parametric vector form for the solution set of a system, which going forward is the required way to write the solution set.

Outline:

Homogeneous and non-homogeneous systems
The solution set of a homogeneous system
The solution set of a non-homogeneous system
The parametric vector form for the solution sets

Video lecture:

Notes from the lecture:

Section 1.5 Solution Sets of Linear Systems (Summer 2020)-1.pdf (updated 5/19 to correct a typo in the last example)