Final Exam Prep Material
This page contains some information/material may be useful in preparing for the final exam. The information here is not exhaustive, and I can pretty much guarantee that there will be at least one question on the final exam this is distinct from any problem that we've done before.
General Study Material
- Course Workbook: The course workbook is a great place to find practice problems. Many of the problems that appear there are previous exam problems
- Linear Algebra and Its Applications by Lay, Lay, and McDonald: Many of the problems we write are similar or based on problems that appear in this textbook.1
Practice Exams
Topics
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
- Addition
- Linear combinations
- Vector equations
- Equivalence with systems of linear equations
- Span
Ax = b
- Matrix-vector multiplication
- Algebraic properties
- Additivity
- Homogeneity
- Algebraic properties
- 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
Footnotes:
1
it's an open secret that it's not difficult to find a pdf copy of this text