We introduced the dot product in Chapter One, but we have not found a lot of use for it so far. In this chapter, we pick it back up and see what we can do with it. Of course, the dot product hides a lot of geometry in its mysteries, but for us the key is the notion of orthogonality.
In this chapter we will address two important questions:
- How can we find approximate solutions to equations \(Ax=b\) when we know (or suspect) that finding a true solutions is impossible?
- How can we use geometry to find a good basis for a subspace?
Now, to begin, we shall explore the idea of subspaces being orthogonal, rather than just vectors, and strengthen the Fundamental Theorem of Linear Algebra.