Harrison Chapman

  1. Teaching
  2. Linear Algebra I
  3. Exam 2 Topics

Exam 2 Topics

Exam 2 will be on Wednesday, March 21 in our normal classroom. No calculators or textbooks will be allowed. You are, however, allowed to bring one handwritten page of notes (front and back) of your own creation to reference (if you have any questions or concerns about what is allowed on this sheet, please send me an email).

There are additional study resources (practice questions and a practice exam) available on Canvas.

In addition to the review problems above, also do review the homework. We’ve had homework 4,5,6 so far. I’d also encourage you to take a look at homeworks 4,5,6,7 from my previous section, although you don’t have to worry about anything mentioning “change of basis”. There are also solutions to most odd-numbered problems in the book.

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.

Topics

  • Material from the last exam that you should still know:
    • 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
    • Column vectors in \(\mathbb R^n\)
    • Adding and scaling column vectors
    • 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
    • Matrix multiplication
    • Matrix properties
    • Inverse matrices
  • New material for Exam 2:
    • Matrix transpose (1.4)
    • Diagonal matrices, triangular matrices, symmetric matrices (1.7)
    • Determinants, meaning, calculations, and properties (no adjoint matrix, no Cramer’s rule) (2.1,2.2,2.3)
    • Relationship between determinants and inverses (Theorem 2.3.8)
    • Basics on Euclidean vector spaces \(\mathbb R^n\) (3.1)
    • Dot products, length, and direction (3.2)
    • Definition of orthogonality (Definition 1 in 3.3)
    • Vector spaces (4.1)
    • Subspaces (4.2)
    • Spans (4.2)
    • Linear independence/dependence (4.3)
    • Definition of a basis (Definition 1 in 4.4)
    • Definition of dimension (up to and including Definition 1 in 4.5)
    • Image (a.k.a. column space) and kernel (a.k.a. null space) (4.7)