MATH 223: Linear Algebra


  • The Final and its solution key are available
  • Solution 11 now posted
  • Solution 10 now posted
  • The office hours of Tuesday, Nov 21, and Friday, Nov 24, are cancelled
  • Typo corrected and an added hint (problem 3) in Assignment 10; Assignment 11 (the last one!) already online
  • Assignment 10 and Solution 9 are available
  • Midterm 2 and its solution key are online
  • Hint added to Problem 3 of Homework 9
  • Assignment 9 posted
  • Solution 8 online
  • Assignment 8 and Solution 7 now posted; Assignment 8 is due on Monday
  • Assignment 7 and Solution 6 now posted
  • Assignment 6 and Solution 5 now posted
  • Assignment 5 now posted
  • Problem 4 in Assignment 4 slightly edited for clarity; update for latest version
  • Midterm 1 and its solution key are online
  • Solution 2 and Homework 3 now posted; Homework 3 due on Monday, Oct. 2nd
  • Solution 1 and Homework 2 now posted
  • Friday office hour moved to 12:00-1:00
  • First Homework Assignment is available
  • Tuesday office hour moved to 8:30-9:30

Basic Information

Homework Assignments

Sheet Number Due Date Solution
Homework 1 Sept. 20th Solution 1
Homework 2 Sept. 27th Solution 2
Homework 3 Oct. 2nd Solution 3
Homework 4 Oct. 11th Solution 4
Homework 5 Oct. 18th Solution 5
Homework 6 Oct. 25th Solution 6
Homework 7 Nov. 1st Solution 7
Homework 8 Nov. 6th Solution 8
Homework 9 Nov. 15th Solution 9
Homework 10 Nov. 22th Solution 10
Homework 11 Nov. 29th Solution 11

Weekly lecture summaries
Week 1 Introduction. Definition of a vector space. Examples (column vectors, matrices, functions). First properties (cancellation law, various uniquenesses,...)
Week 2 Definition of a subspace. Examples. Linear combinations. Definition of a basis. Spanning sets and linearly independent sets.
Week 3 Extraction of a basis from a spanning set. Definition of the dimension. The replacement lemma. Bases and dimension. Cardinality of spannning and linearly independent sets. Subspaces of finite dimensional spaces.
Week 4 Defintion of a linear transformation. Elementary properties. Examples. The set of linear maps as a vector space. Range and kernel. The dimension theorem. Example.
Week 5 p x n matrices. The matrix product. Properties. The matrix product is not commutative. Examples. Traces. Invertible matrices. The matrix of a linear transformation w.r.t. two bases.
Week 6 Linear transformations and matrices: Examples. The composition of linear maps and the matrix product. Change of bases.
Week 7 Systems of linear equations. Coefficients and augmented matrices. Row reduction. The echelon and reduced echelon forms. Consequences: Existence and uniqueness of solutions. The rank of a matrix.
Week 8 More on row reduction. Examples. Invertibility & existence and uniqueness of solutions. The general solution of inhomogeneous systems. The determinant: introduction and first properties.
Week 9 The determinant and invertibility. Computing the determinant by column reduction. The determinant of a product. Existence: The cofactor expansion. Examples. Eigenvalues and eigenvectors: definition and first examples.
Week 10 Diagonalizable linear maps and matrices. Characterization of the eigenvalues. The characteristic polynomial and its roots. Computing eigenvalues. Eigenspaces. Algebraic and geometric multiplicities, and relation to diagonalizability.
Week 11 More on diagonalization, examples. Inner product spaces: Introduction and definitions. The norm. Examples. The Cauchy-Schwarz inequality.
Week 12 Proof of Cauchy-Schwarz. The triangle inequality and Pythagoras' theorem. Orthonormal bases. The adjoint of a linear map. Self-adjoint linear transformations and matrices. The spectral theorem for self-adjont matrices. Google Page Rank.
Week 13 Recap: inner products, norms, adjoints of maps and matrices. Orthogonal and unitary transformations. The spectral theorem rephrased. Examples. General overview of the course.