# Harrison Chapman

## Exam 2 Topics

Exam 2 will be on Wednesday, November 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 6, 7, 8, and 9. 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.

You may find questions 1, 2, 3, 4, 5, 7, 8, and 9 on the practice exam here (solutions here) useful.

• Chapter 9:
• Subspaces of a vector space
• Checking whether a set of vectors is a subspace or not
• Span notation, and the span of a set of vectors
• How to know if a vector is in a span of other vectors
• Chapter 10:
• Linear independence
• Linear dependence
• Checking if a set of vectors is linearly dependent or not
• Checking if a set of $n$ $n$-column vectors is linearly dependent or not
• Chapter 11:
• Basis
• Dimension
• How to find the dimension of a span of a set of vectors
• How to change a $B$ vector to a $C$ vector (if $B$, $C$ are bases)
• How to change a square matrix for a linear transformation for basis $B$ to one for a basis $C$
• Bases for vector spaces of weirder type (e.g. matrices, polynomials)
• Chapter 12:
• The eigenvector equation
• Eigenvalues; what they are and how to find them
• Eigenvectors; what they are and how to find them
• Eigenspaces; what they are and how to find them
• Characteristic polynomial for a matrix M, how to compute, how to use
• How many eigenvalues are there for a matrix if we allow real entries? Complex entries?
• Chapter 13:
• Diagonalizability of square matrices
• When is a matrix diagonalizable?
• What is a diagonalizable matrix?
• What are the advantages of a diagonalizable matrix?
• How can you diagonalize a given matrix?
• Under which bases do matrices diagonalize?