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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:

  1. How can we find approximate solutions to equations \(Ax=b\) when we know (or suspect) that finding a true solutions is impossible?
  2. How can we use geometry to find a good basis for a subspace?
The answers to both of these questions will involve using the dot product to check for orthogonality. This will be leveraged into the technique of orthogonal projection.

Now, to begin, we shall explore the idea of subspaces being orthogonal, rather than just vectors, and strengthen the Fundamental Theorem of Linear Algebra.