Schedule

TUE 09-03 Lecture 1

• Title: Linear Equations
  
• Topics:
  
  • Introduce systems of linear equations and discuss how to solve them
  • Learn how to use matrices to represent systems of linear equations
• Material:
  
  • Reading:
    
    • Linear Algebra, Geomtry, and Computation (LAGC): Linear Equations
      
    • Linear Algebra @BUCS (LABU): Linear Equations
      
    • Interactive Linear Algebra (ILA) 1.1: Systems of Linear Equations
      
  • Slides:
    
    • keynote
      
    • pdf
      
    • pdf (A1 annotations)
      
    • pdf (A2 annotations)
      
  • Administrivia:
    
    • keynote
      
    • pdf
      
• Reminders:
  
  • read course information and course policies
  • install Python + libraries
  • get access to Piazza and Gradescope
  • No office hours this week

THU 09-05 Lecture 2

• Title: Solving Linear Systems
  
• Topics:
  
  • Continue looking at solve systems of linear equations
  • Examine the relationship between row operations and solving systems of linear equations
  • See how to use SymPy to solve linear systems, both step by step, and in a single function call
• Material:
  
  • Reading:
    
    • LAGC: Linear Equations
      
    • LABU: Matrices and Row Operations
      
    • ILA 1.2: Row Reductions
      
    • Sympy: Matrices
      
  • Slides:
    
    • keynote
      
    • pdf
      
  • Homework 1:
    
    • html
      
• Reminders:
  
  • Homework 1 is assigned

FRI/MON 09-06/09 Discussion 1

• Topics:
  
  • Set up your Python environment (with help from the TF/TAs)
  • Practice with SymPy
• Material:
  
  • Installation Guide
    

TUE 09-10 Lecture 3

• Title: Echelon Forms
  
• Topics:
  
  • Understand the shape of matrices which represent solutions to a system of linear equations (we need to know how to read off a solution from the output of Gaussian elimination)
• Material:
  
  • Reading:
    
    • LAGC: Gaussian Elimination
      
    • LABU: Echelon Forms
      
  • Slides:
    
    • keynote
      
    • pdf
      
    • pdf (A1 annotations)
      
    • pdf (A2 annotations)
      

THU 09-12 Lecture 4

• Title: Gaussian Elimination + Numerics
  
• Topics:
  
  • Introduce and analyze Gaussian elimination, an algorithm for solving general systems of linear equations
  • Revisit the problem of solving systems of linear equations with computers, this time in the presence of floating-point error
  • Look more deeply at NumPy and how it can be used to reason about the concepts we've seen so far
• Material:
  
  • Reading:
    
    • LAGC: Gaussian Elimination
      
    • LAGC: Numerics
      
    • LABU: Gaussian Elimination
      
    • LABU: Supplementary NumPy Tutorial
      
    • Numpy: the absolute basics for beginners
      
    • Floating-Point Arithemtic: Issues and Limitations
      
  • Slides:
    
    • keynote
      
    • pdf
      
    • pdf (A1 annotations)
      
    • pdf (A2 annotations)
      
  • IEEE 754 Converter
    
  • Homework 2:
    
    • html
      
    • hw02-starter.zip
      
• Reminders:
  
  • Homework 1 is due
  • Homework 2 is assigned

FRI/MON 09-13/16 Discussion 2

• Topics: Practice solving linear systems
  

TUE 09-17 Lecture 5

• Title: Vector Equations
  
• Topics:
  
  • Connect the algebraic notion of a linear equation to the geometric notation of a vectors
  • See how geometric properties of vectors reduce to solving systems of linear equations
  • Look at the algebra of vectors
• Material:
  
  • Reading:
    
    • LAGC: Vector Equations
      
    • ILA 2.1: Vectors
      
    • ILA 2.2: Vector Equations and Spans
      
  • Slides:
    
• Reminders:
  

THU 09-19 Lecture 6

• Title: Matrix-Vector Equations
  
• Topics:
  
  • Define matrix-vector multiplication
  • Use matrices-vector equations to represent systems of linear equations
  • Look at the algebra of matrices and vectors
• Material:
  
  • Reading:
    
    • LAGC: Ax = b
      
    • ILA 2.3: Matrix Equations
      
  • Slides:
    
  • Homework 3:
    
• Reminders:
  
  • Homework 2 is due
  • Homework 3 is assigned

FRI/MON 09-20/23 Discussion 3

• Topics:
  
  • Practice with vectors, spans, and linear independence
  • Practice with NumPy (compare to SymPy)
• Material:
  
• Reminders:
  

TUE 09-24 Lecture 7

• Title: Linear Independence
  
• Topics:
  
  • Introduce notion of linear independence as a way to understand if the span of a set of vectors is "as large as possible"
  • Examine the relationship between linear independence and systems of linear equations (in particular, look at what the shape of a matrix says about the linear dependences of its columns)
• Material:
  
  • Reading:
    
    • LAGC: Linear Independence
      
    • ILA 2.5: Linear Independence
      
  • Slides:
    
• Reminders:
  

THU 09-26 Lecture 8

• Title: Linear Transformations
  
• Topics:
  
  • Introduce linearity as a way of describing "well-behaved" functions on vectors.
  • See examples and non-examples of linear functions (in particular, look at matrix-vector multiplication as the canonical example of a linear transformation)
• Material:
  
  • Reading:
    
    • LAGC: Linear Transformations
      
    • ILA 3.1: Matrix Transformations
      
    • ILA 3.3: Linear Transformations
      
  • Slides:
    
  • Homework 4:
    
• Reminders:
  
  • Homework 3 is due
  • Homework 4 is assigned

FRI/MON 09-27/30 Discussion 4

• Topics:
  
  • Practice with linear transformations
• Material:
  
• Reminders:
  

TUE 10-01 Lecture 9

• Title: Matrices of Linear Transformations
  
• Topics:
  
  • Show that, in fact, every linear transformation can be represented as a matrix transformation
  • Introduce the notion of a basis (the simplest example being the standard basis) as a way of "decomposing" vectors simpler parts
• Material:
  
  • Reading:
    
    • LAGC: Matrix of a Linear Transformation
      
    • ILA 3.2: One-to-one and Onto Transformations
      
    • ILA 3.3: Linear Transformations
      
  • Slides:
    
• Reminders:
  

THU 10-03 Lecture 10

• Title: Matrix Algebra
  
• Topics:
  
  • Define matrix multiplication and look at how it interacts with other matrix operations like addition and scalar multiplication
  • Connection matrix multiplication with the composition of linear transformations
• Material:
  
  • Reading:
    
    • LAGC: Matrix Algebra
      
    • ILA 3.4: Matrix Multiplication
      
  • Slides:
    
  • Homework 5:
    
• Reminders:
  
  • Homework 4 is due
  • Homework 5 is assigned

FRI/MON 10-04/07 Discussion 5

• Topics:
  
  • Practice with matrix inversion
  • More Practice with NumPy
• Material:
  
• Reminders:
  

TUE 10-08 Lecture 11

• Title: Matrix Inverse
  
• Topics:
  
  • Show how to "divide" by a matrix as a way of solving systems of linear equations (when possible)
  • Examine how matrix inversion interacts with other matrix operations
• Material:
  
  • Reading:
    
    • LAGC: The Inverse of a Matrix
      
    • ILA 3.5: Matrix Inverses
      
    • ILA 3.6: The Invertible Matrix Theorem
      
  • Slides:
    
• Reminders:
  

THU 10-10 Lecture 12

• Title: Invertible Matrix Theorem
  
• Topics:
  
  • Review matrix inversion
  • Connect everything we've seen so far in the case of square matrices (one theorem to rule them all…)
• Material:
  
  • Reading:
    
    • LAGC: The Inverse of a Matrix
      
    • ILA 3.6: The Invertible Matrix Theorem
      
  • Slides:
    
  • Homework 6:
    
• Reminders:
  
  • Homework 5 is due
  • Homework 6 is assigned

FRI/TUE 10-11/15 Discussion 6

• Topics:
  
  • More practice with matrix inversion
  • More practice with NumPy
• Material:
  
• Reminders:
  
  • The discussion section normally held on Monday will be held on Tuesday because of Indigenous People's Day

THU 10-17 Lecture 13

• Title: Algebraic Graph Theory
  
• Topics:
  
  • See how matrices can be used to represent graphs (a.k.a. networks)
  • See how matrix operations can be interpreted as operations on graphs
  • Set up the conceptual framework for Markov chains and PageRank
• Material:
  
  • Reading:
    
    • TODO Algebraic Graph Theory
      
  • Slides:
    
• Reminders:
  
  • Homework 6 is due
  • There is no class on Tuesday this week because of Indigenous People's Day

FRI/MON 10-18/21 Discussion 7

• Topics:
  
  • Practice with adjacency matrices
  • Examples with NetworkX
• Material:
  
• Reminders:
  

TUE 10-22 Midterm Exam

THU 10-24 Lecture 14

• Title: Markov Chains
  
• Topics:
  
  • Introduce Markov chains as an application of the topics we've covered
  • Use Markov chains to reason about the long-term behavior of linear dynamical systems
• Material:
  
  • Reading:
    
    • LAGC: Markov Chains
      
  • Slides:
    
  • Homework 7:
    
• Reminders:
  
  • Homework 7 is assigned

FRI/MON 10-25/28 Discussion 8

• Topics:
  
  • Practice with Markov Chains
• Material:
  
• Reminders:
  

TUE 10-29 Lecture 15

• Title: Matrix Factoriazations
  
• Topics:
  
  • Discuss matrix factorization in general as a way to "get more information" about a matrix
  • Look at the LU factorization as a faster way of solving the multiple matrix equations (over the same matrix)
• Material:
  
  • Reading:
    
    • LAGC: Matrix Factorizations
      
  • Slides:
    
• Reminders:
  

THU 10-31 Lecture 16

• Title: Computer Graphics
  
• Topics:
  
  • Switch gears to talk about linear algebra and computer graphics, in particular the use of linear transformations and perspective transformations for rendering images of 3D objects on a 2D screen
• Material:
  
  • Reading:
    
    • LAGC: Computer Graphics
      
  • Slides:
    
  • Homework 8:
    
• Reminders:
  
  • Homework 7 is due
  • Homework 8 is assigned

FRI/MON 11-01/04 Dicussion 9

• Topics:
  
  • Help with Homework 8
• Material:
  
• Reminders:
  

TUE 11-05 Lecture 17

• Title: Subspaces
  
• Topics:
  
  • Introduce subspaces and bases as a way to think more generally about space
  • Extend our intuitions about planes in R³ to subspaces in Rⁿ
  • Connect subspaces to matrices and solving systems of linear equations
• Material:
  
  • Reading:
    
    • LAGC: Subspaces
      
    • ILA 2.6: Subspaces
      
  • Slides:
    
• Reminders:
  

THU 11-07 Lecture 18

• Title: Dimension and Rank
  
• Topics:
  
  • Introduce dimension as a way of quantifying how "large" a span is
  • Learn techniques for finding bases for the column space and the null space of a matrix
  • Relate the dimension of the column space and the null space of a matrix
• Material:
  
  • Reading:
    
    • LAGC: Dimension and Rank
      
    • ILA 2.7: Basis and Dimension
      
    • ILA 2.9: The Rank Theorem
      
  • Slides:
    
  • Homework 9:
    
• Reminders:
  
  • Homework 8 is due
  • Homework 9 is assigned

FRI/MON 11-08/11 Discussion 10

• Topics:
  
  • Practice with the rank theorem
  • Practice with dimension and rank in NumPy
• Material:
  
• Reminders:
  

TUE 11-12 Lecture 19

• Title: Eigenvalues and Eigenvectors
  
• Topics:
  
  • Introduce eigenvectors as as way of thinking about vectors which are "just stretched" by a matrix
  • Determine how to verify eigenvectors and eigenvalues of a matrix
  • Use eigenvectors to reason about linear dynamical systems
• Material:
  
  • Reading:
    
    • LAGC: Eigenvectors and Eigenvalues
      
    • ILA 5.1: Eigenvectors and Eigenvalues
      
  • Slides:
    
• Reminders:
  

THU 11-14 Lecture 20

• Title: The Characteristic Equation
  
• Topics:
  
  • Look breifly at the notion of the determinant of a matrix
  • Determine how to find eigenvalues (not just verify them)
  • Connect eigenvalues to polynomials (?)
• Material:
  
  • Reading:
    
    • LAGC: The Characteristic Equation
      
    • ILA 5.2: The Characteristic Polynomial
      
  • Slides:
    
  • Homework 10:
    
• Reminders:
  
  • Homework 9 is due
  • Homework 10 is assigned

FRI/MON 11-15/18 Discussion 11

• Topics:
  
  • Practice with eigenvalues and eigenvectors
  • More practice with NumPy
• Material:
  
• Reminders:
  

TUE 11-19 Lecture 21

• Title: Diagonalization
  
• Topics:
  
  • Examine another matrix factorization related to basis changes
  • Walk through how to diagonlize a matrix
• Material:
  
  • Reading:
    
    • LAGD: Diagonalization
      
    • ILA 5.4: Diagonalization
      
  • Slides:
    
• Reminders:
  

THU 11-21 Lecture 22

• Title: Orthogonality
  
• Topics:
  
  • Introduce familiar notions like "length" and "angles" into our study of vectors
  • Look at the special case of "right" angles between vectors, i.e., orthogonality
• Material:
  
  • Reading:
    
    • LAGC: Analytic Geometry in Rⁿ
      
    • ILA 6.1: Dot Products and Orthogonality
      
  • Slides:
    
  • Homework 11:
    
• Reminders:
  
  • Homework 10 is due
  • Homework 11 is assigned

FRI/MON 11-22/25 Discussion 12

• Topics:
  
  • Practice with diagonalization
  • Practice with diagonalization in NumPy
• Material:
  
• Reminders:
  

TUE 11-26 Lecture 23

• Title: PageRank
  
• Topics:
  
  • Connect random walks with eigenvectors
  • Discuss PageRank and Google search from the perspective of Markov chains
• Material:
  
  • Reading:
    
    • LAGC: PageRank
      
    • The anatomy of a large-scale hypertextual Web search engine
      
  • Slides:
    
• Reminders:
  
  • There is no class on Thursday because of the Thanksgiving recess

TUE 12-03 Lecture 24

• Title: Orthogonal Projection
  
• Topics:
  
  • Introduce orthogonal projection as a way of finding the "shadow" of a vector in a subspace
  • Connect orthogonality to matrices and linear transformations
• Material:
  
  • Reading:
    
    • LAGC: Orthogonal Sets and Projections
      
    • ILA 6.3: Orthogonal Projections
      
  • Slides:
    
• Reminders:
  
  • There will be no discussion section on Monday because of the Thanksgiving recess

THU 12-05 Lecture 25

• Title: Least Squares
  
• Topics:
  
  • Introduce the least squares method as a way of giving "approximate" solutions to systems of linear equations
  • Demonstrate the relationship between orthogonal projection and least squares solutions
• Material:
  
  • Reading:
    
    • LAGC: Least Squares
      
    • ILA 6.5: The Method of Least Squares
      
  • Slides:
    
  • Homework 12:
    
• Reminders:
  
  • Homework 11 is due
  • Homework 12 is assigned (it is shorter than usual)

FRI/MON 12-06/09 Discussion 13

• Topics:
  
  • Practice with least squares
  • Practice with least squares in NumPy
• Material:
  
• Reminders:
  

TUE 12-10 Lecture 26

• Title: Linear Models
  
• Topics:
  
  • Look at applications of linear algebra in machine learning
• Material:
  
  • Reading:
    
    • LAGC: Linear Models
      
  • Slides:
    
• Reminders:
  
  • Homework 12 is due
  • Last day of class (!)

??? ??-?? Final Exam