2: Systems of Linear Equations- Geometry

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We have already discussed systems of linear equations and how this is related to matrices. In this chapter we will learn how to write a system of linear equations succinctly as a matrix equation, which looks like \(Ax=b\text{,}\) where \(A\) is an \(m \times n\) matrix, \(b\) is a vector in \(\mathbb{R}^m\) and \(x\) is a variable vector in \(\mathbb{R}^n\). As we will see, this is a powerful perspective. We will study two related questions:

What is the set of solutions to \(Ax=b\text{?}\)
What is the set of \(b\) so that \(Ax=b\) is consistent?

The first question is the kind you are used to from your first algebra class: what is the set of solutions to \(x^2-1=0\). The second is also something you could have studied in your previous algebra classes: for which \(b\) does \(x^2=b\) have a solution? This question is more subtle at first glance, but you can solve it in the same way as the first question, with the quadratic formula.

In order to answer the two questions listed above, we will use geometry. This will be analogous to how you used parabolas in order to understand the solutions to a quadratic equation in one variable. Specifically, this chapter is devoted to the geometric study of two objects:

the solution set of a matrix equation \(Ax=b\text{,}\) and
the set of all \(b\) that makes a particular system consistent.

The second object will be called the column space of \(A\). The two objects are related in a beautiful way by the rank theorem in Section 2.8.

Instead of parabolas and hyperbolas, our geometric objects are subspaces, such as lines and planes. Our geometric objects will be something like 13-dimensional planes in \(\mathbb{R}^{27}\text{,}\) etc. It is amazing that we can say anything substantive about objects that we cannot directly visualize.

We will develop a large amount of vocabulary that we will use to describe the above objects: vectors (Section 2.1), spans (Section 2.2), linear independence (Section 2.5), subspaces (Section 2.6), dimension (Section 2.7), coordinate systems (Section 2.8), etc. We will use these concepts to give a precise geometric description of the solution set of any system of equations (Section 2.4). We will also learn how to express systems of equations more simply using matrix equations (Section 2.3).

2.1: Vectors
We have been drawing points in Rⁿ as dots in the line, plane, space, etc. We can also draw them as arrows. Since we have two geometric interpretations in mind, we now discuss the relationship between the two points of view.
2.2: Vector Equations and Spans
The thing we really care about is solving systems of linear equations, not solving vector equations. The whole point of vector equations is that they give us a different, and more geometric, way of viewing systems of linear equations.
2.3: Matrix Equations
In this section we introduce a very concise way of writing a system of linear equations: Ax=b. Here A is a matrix and x,b are vectors (generally of different sizes).
2.4: Solution Sets
In this section we will study the geometry of the solution set of any matrix equation Ax=b.
2.5: Linear Independence
Sometimes the span of a set of vectors is “smaller” than you expect from the number of vectors, as in the picture below. This means that (at least) one of the vectors is redundant: it can be removed without affecting the span. In the present section, we formalize this idea in the notion of linear independence.
2.6: Subspaces
2.7: Basis and Dimension
2.9: The Rank Theorem
2.8: Bases as Coordinate Systems