# Harrison Chapman

## Exam 1 Topics

Exam 1 will be on Wednesday, October 4 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 review problems listed below, also do review the homework. 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.

• Chapter 2:
• Systems of linear equations
• Gaussian elimination
• Row echelon and Reduced row echelon forms
• Using pivots of (reduced) row echelon matrices to express solutions and how many solutions a system of linear equations has.
• Row operations and elementary matrices
• Some extra review problems: 2.2.5, 2.2.9, 2.6.1, 2.6.4
• Chapter 4:
• Euclidean vector spaces $\mathbb R^n$
• Column vectors in $\mathbb R^n$
• Adding and scaling column vectors
• Lines, planes, hyperplanes
• Euclidean magnitude (length/distance)
• Dot products and angles between vectors
• Some extra review problems: 4.5.2, 4.5.5, 4.5.10
• Chapter 5:
• Vector spaces and the ten vector space axioms
• Examples and counterexamples
• Some extra review problems: 5.3.1, 5.3.2a, 5.3.6
• Chapter 6:
• Linear transformations: $L(r\vec u + s\vec v) = rL(\vec u) + sL(\vec v)$
• Is a function linear or not?
• Expressing linear transformations as matrices
• Some extra review problems: 6.5.2, 6.5.3, 6.5.4
• Chapter 7:
• Every vector can be expressed as a linear combination of basis vectors
• Matrix multiplication
• Matrix properties
• Inverse matrices
• Some extra review problems: 7.2.3e, 7.4.1, 7.4.3, 7.4.4, 7.4.7, 7.6.5, 7.6.6
• Chapter 8:
• Determinants
• Determinants by expansion formula
• Determinants of elementary matrices
• Determinants by Gaussian elimination
• Properties of the determinant
• Classical adjoints and a formula for the inverse of invertible matrices
• Some extra review problems: 8.3.5, 8.3.11, 8.5.1 (but use Gaussian elimination instead), 8.5.2