Harrison Chapman

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

Exam 3 Topics

Exam 3 will be on Wednesday, April 25 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).

In addition to the review problems above, also do review the homework. We’ve had homework 7,8,9 so far. I’d also encourage you to take a look at the homework from my previous section. 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 exams 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
    • 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)
  • New material for Exam 3
    • Eigenvalues (5.1)
    • Eigenvectors (5.1)
    • Eigenspaces (5.1)
    • Bases for eigenspaces (5.1)
    • Bases of eigenvectors (5.2)
    • Diagonalization: “\(A = PDP^{-1}\)” (5.2)
    • When a matrix is diagonalizeable or not (5.2)
    • There are always \(n\) complex eigenvalues of an \(n \times n\) matrix (5.3)
    • Complex eigenvalues always come in pairs (5.3)
    • What is an inner product (6.1)
    • Dot product is an inner product (6.1)
    • Magnitude an inner product (6.1)
    • Orthogonality (6.2)
    • Orthogonal matrices (6.2)
    • Orthogonal and orthonormal bases (6.2)
    • Gram-Schmidt Process (6.3)
    • Rank (= dimension of column space) and nullity (= dimension of kernel) (4.8)
    • Rank+nullity = #rows (4.8)