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Final Exam Prep Material

Practice Exams

Topics

This is a list of topics that may appear on the final exam. Generally speaking, anything in our textbook is fair game, and this list is not guaranteed to be exhaustive. The primary aim of this list is to point out a couple topics that will not appear on the final.

Systems of linear equations

  • Counting solutions
    • Consistency
    • Uniqueness
    • Infinite solutions
  • Coefficient matrices
  • Augmented matrices
  • Finding solutions
    • Elimination
    • Back substitution
  • Verifying solutions
  • Elementary row operations
    • Replacement
    • Interchange
    • Scaling
  • Row Equivalence

Gaussian elimination

  • Echelon forms
  • Reduced echelon forms
  • Pivot positions and pivot rows
  • Performing Gaussian elimination
    • Elimination
    • Back substitution
  • Interpreting echelon forms
    • Basic variables
    • Free variables
    • General-form solutions
  • EXCLUDING: FLOP counts

Vector equations

  • (Column) vectors
  • Vector operations
    • Addition
      • Parallelogram rule
      • Tip-to-tail rule
    • Multiplication
    • Scaling
    • Algebraic properties
  • Linear combinations
  • Vector equations
    • Equivalence with systems of linear equations
  • Span

Ax = b

  • Matrix-vector multiplication
    • Algebraic properties
      • Additivity
      • Homogeneity
  • The matrix equation
    • Equivalence with systems of linear equations
  • Inner products
  • The identity matrix

Linear independence

  • Linear dependence
  • Spanning sets
  • Network flow

Linear transforms

  • Definitions
    • Domain
    • Codomain
    • Image
    • Range
  • Linearity
  • Algebraic properties (of Linear Transformations)
  • Simple linear transformations
    • Shearing
    • Contraction
    • Dilation
    • Rotation
    • Reflection
    • Projection
  • Simple non-linear transformations
    • Translation
  • Non-geometric word problems
    • Manufacturing example

Matrix of a transformation

  • Finding the matrix implementing a linear transformation
  • Kinds of linear transformations
    • One-to-one
    • Onto
  • 2 × 2 Determinants
    • EXCLUDING: Relationship with area

Matrix algebra

  • Matrix Operations
    • Addition
    • Multiplication
      • Relation to composition
    • Scaling
    • Algebraic properties
      • Matrix multiplication is not commutative
    • Powers
    • Transposition
  • Inner product
  • EXCLUDING: Computational viewpoint

Matrix inverses

  • Invertibility
  • Singular vs. non-singular matrices
  • Determinants
  • Equivalence to solving systems of linear equations
  • EXCLUDING:
    • Computational view
    • Ill-conditioned matrices
  • The invertible matrix theorem

Markov chains

  • Linear dynamical systems/linear difference equations
  • Probability vectors
  • Stochastic matrices
  • Steady-states
  • Convergence to steady states

Matrix factorization

  • LU factorization
  • Elementary matrices
  • EXCLUDING:
    • Forward substitution
    • FLOP counts
    • Pivoting

Computer graphics

  • Homogeneous coordinates
  • Rotation matrices
  • Perspective projections
  • Composing transformation
  • EXCLUDING: programming aspects of graphics

Subspaces

  • Equivalence with spans
  • Column space
  • Null space
  • Basis
    • of column space
    • of null space

Dimension and rank

  • Coordinate systems
  • Dimension of a subspace
    • of column space
    • of null space
  • Rank
  • Rank-nullity theorem
  • EXCLUDING: Isomorphism

Eigenvectors and eigenvalues

  • Checking eigenvectors/values
  • Finding eigenvectors
  • Eigenspace
  • Eigenvalues of triangular matrices
  • Invertibility and eigenvalues
  • Eigenvalues solve difference equations

The characteristic equation

  • Determinants
  • Finding the characteristic polynomial
  • Finding eigenvalues
  • Similarity
  • Complete solution for a Markov chain

Diagonalizability

  • Diagonal matrices
  • Similar Matrices
  • Diagonalizability
  • Eigenbasis
  • Exposing the behavior of dynamical systems

Analytic geometry

  • Inner product/dot product
  • Norm
  • Distance
  • Angle
  • Orthogonality
  • Cosine similarity

Orthogonal projection

  • Orthogonal basis
  • Orthogonal projection
  • Projections and coordinates
  • Orthonormal sets
  • Orthogonal matrices

Least squares

  • Finding least squares
  • Normal equations
  • Projecting onto a basis

Linear models

  • Best fit line/quadratic
  • Multiple regression
  • Design matrices

Symmetric matrices

  • Orthogonal diagonalization
  • Quadratic forms
  • Definiteness
  • Constrained optimization

Singular value decomposition

  • \(\|A\mathbf x\|^2\) as a quadratic form
  • Singular values
  • Determining the SVD of a matrix
  • Reduced SVD and the psuedoinverse
  • EXCLUDING: Applications of the SVD