Harrison Chapman

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

Final Exam Topics

Our final exam will be on Thursday, May 10 from 4:10pm–6:10pm 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).

Topics

  • Unit 1: Linear transformations and matrices
    • Systems of linear equations (1.1)
    • Gaussian elimination (1.2)
    • Row echelon and Reduced row echelon forms (1.2)
    • Using pivots of (reduced) row echelon matrices to express solutions and how many solutions a system of linear equations has. (1.2)
    • Row operations and elementary matrices (1.2)
    • Column vectors in \(\mathbb R^n\) (1.3)
    • Adding and scaling column vectors (1.3)
    • Linear transformations: \(L(r\vec u + s\vec v) = rL(\vec u) + sL(\vec v)\) (1.8 (10e:4.9))
    • Is a function linear or not? (1.8 (10e:4.9))
    • Expressing linear transformations as matrices (1.8 (10e:4.9))
    • Matrix multiplication (1.3,1.4,1.5)
    • Matrix properties (1.3,1.4,1.7)
    • Inverse matrices (1.4,1.5)
  • Unit 2: Linear independence, bases, and subspaces
    • 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)
  • Unit 3: Diagonalization and orthogonality
    • 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 = #columns (4.8)