# Harrison Chapman

## Exam 3 Topics

Exam 3 will be on Friday, December 1 in our normal classroom. No calculators or textbooks will be allowed. You are, however, allowed to bring one page of notes (front and back) of your own creation to reference.

In addition to the topics below review the homework, specifically assignments 10, 11, and 12. This is not guaranteed to be an absolute list of all topics covered so far, but I hope it may help you to guide your study.

• Invertibility of square matrices:

We talked a good deal this unit about many different equivalent criteria for matrices to be invertible. I suggest you study these well.

• Chapter 14:
• Inner products
• Unit vectors
• Orthogonal bases
• Orthonormal bases
• Orthogonal matrices
• Inner products are dot products (if you have an orthonormal basis)
• Change of basis between orthonormal bases
• Orthogonal decompositions of vectors
• Gram-Schmidt algorithm for orthogonalization (orthonormalization)
• Orthogonal complements to subspaces
• Chapter 16:
• Range of a transformation (and how to find bases)
• Image of a subset under a transformation
• Pre-image of a subset under a transformation
• Rank of a transformation
• Row and column space of a matrix (and how to find bases)
• One-to-one (injective), onto (surjective), and bijective transformations
• Kernel of a transformation (and how to find bases)
• dim(V) = rank(L) + null(L)
• Criteria for invertibility (see note above)
• Chapter 17:
• Least squares solutions
• Projection matrices