APMA 0260

Welcome to APMA 0260 – Linear Algebra and Multivariable Calculus for Applied Mathematicians! This is the course to give you a head-start to the Applied Mathematics concentration. On this website, you can find the syllabus, lecture notes, homework assignments, and study materials.

Syllabus

APMA 0260 – Syllabus (Spring 2025)

Homework Extension Request Form

Lecture Notes

Note: The lecture notes will be frequently updated. Please refresh your webpage so that you have the most up-to-date version of the notes

#
Date
Lecture Title
 YouTube Videos
1 Jan 22 Systems of Equations Inconsistent Systems
2 Jan 24 Row-reduction Row-reduction
Infinitely Many Solutions
Solving ax + by
3 Jan 27 Vectors
4 Jan 29 Span Span
Row Theorem
Particular Solution
5 Jan 31 Linear Independence Linear Independence
6 Feb 3 Linear Transformations (I) Linear Transformations
7 Feb 5 Linear Transformations (II) Rotation Matrix
One-to-one and onto
8 Feb 7 Matrix Algebra Matrix Multiplication
AB vs BA
Magic Matrix (optional)
9 Feb 10 The Inverse of a Matrix Finding A-1
Product of elementary matrices (optional)
LU Factorization (optional)
10  Feb 12 Subspaces Show A is invertible
Invertible matrices must be square
Show H is a subspace
Show H is not a subspace
Span is a subspace
Vector Space Examples (optional)
11 Feb 14 Nul(A) and Col(A) Nul(A)
Nul(A) is a subspace
Checking something is a basis
12 Feb 19 Dimension and Rank Basis and dimension
Nul(A), Col(A), Rank(A)
Coordinates
Coordinates Matrix
Infinite-dimensional space (optional)
Rank 1 Matrices (optional)
13 Feb 21 Determinants Determinants
Determinants and row-reduction
Vandermonde Determinant
Determinants and Invertibility
Determinant Battle
14 Feb 24 Applications of Determinants Cramer’s Rule
A-1 using determinants
Area of parallelogram
Determinants and volumes
15 Feb 26 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors
How to find Eigenvalues
How to find Eigenvectors
16 Feb 28 Diagonalization Diagonalize 2×2 matrix
Diagonalize 3×3 matrix
Diagonalization analogy
Diagonalization
17 Mar 3 Midterm 1 – Review 111 True/False Questions
Mar 5 Midterm 1
18 Mar 7 Applications of Diagonalization Pokémon Battle
Fibonacci Sequence
Matrix Limit 1
Matrix Limit 2
Square root of a matrix
Matrix exponential
Cos of a matrix
2 to a matrix
Systems of ODE (optional)
Imaginary Root of a Matrix (optional)
Matrixth Root of a Matrix (optional)
Matrix choose a Matrix (optional)
19 Mar 10 Dot Product and Lengths Google and Eigenvalues
20 Mar 12 Orthogonality Awesomeness Orthogonal Sets
Orthogonality
21 Mar 14 Orthogonal Projections Orthogonal Projections
Proof of Cauchy-Schwarz
Awesome Cauchy-Schwarz (optional)
22 Mar 17 Functions of Several Variables Kamehameha Equation
23 Mar 19 Partial Derivatives What is a partial derivative?
Calculating Partial Derivatives
fxy ≠ fyx (optional)
Chain Rule Surprise
Implicit Differentiation
What is a PDE? (optional)
Solutions of PDE
24 Mar 21 Tangent Planes Tangent Planes
Linear Approximations
25 Mar 31 The Chain Rule The Chain Rule
26 Apr 2 Gradients and Applications Sum of Intercepts
27 Apr 4 Max and Min Values The True Second-Derivative Test
IKEA Problem
28 Apr 7 Lagrange Multipliers Another IKEA Problem
Distance Point and Plane
29 Apr 9 Double Integrals (I) What is a Double Integral
Volume of Ravioli
Fubini Counterexample (optional)
30 Apr 11 Double Integrals (II) Changing Order of Integration
Multivariable Integral
31 Apr 14 Midterm 2 – Review Square root of a matrix
111 True/False Questions
Apr 16 Midterm 2
32 Apr 18 Triple Integrals Triple Integrals
Volume of Gouda Cheese
Integral over Cannoli
Volume of two cylinders
Volume of three cylinders (optional)
33 Apr 21 Change of Variables (I) The Jacobian 1
34 Apr 23 Change of Variables (II)
35 Apr 25 Final Exam – Review
May 14 Final Exam

 

Homework Assignments

Homework
Due Date
Solutions
Homework 1 Jan 31 Solutions
Homework 2 Feb 7 Solutions
Homework 3 Feb 14 Solutions
Homework 4 Feb 21 Solutions
Homework 5 Feb 28 Solutions
Homework 6 Mar 14 Solutions
Homework 7 Mar 21 Solutions
Homework 8 Apr 4 Solutions
Homework 9 Apr 11 Solutions
Homework 10 Apr 25
Homework 11 May 2 Optional
Mini Project May 2

Worksheets

Worksheet
Date
Solutions
Worksheet 1 Jan 30 Solutions
Worksheet 2 Feb 6 Solutions
Worksheet 3 Feb 13 Solutions
Worksheet 4 Feb 20 Solutions
Worksheet 5 Feb 27 Solutions
Worksheet 6 Mar 13 Solutions
Worksheet 7 Mar 20 Solutions
Worksheet 8 Apr 3 Solutions
Worksheet 9 Apr 10 Solutions
Worksheet 10 Apr 24

Quizzes

Quiz
Date
Solutions
Quiz 1 Jan 30 Solutions
Quiz 2 Feb 6 Solutions
Quiz 3 Feb 13 Solutions
Quiz 4 Feb 20 Solutions
Quiz 5 Feb 27 Solutions
Quiz 6 Mar 13 Solutions
Quiz 7 Mar 20 Solutions
Quiz 8 Apr 3 Solutions
Quiz 9 Apr 10 Solutions
Quiz 10 Apr 24

YouTube Playlists

Chapter Playlist
1 Linear Equations in Linear Algebra
2 Matrix Algebra
3 Determinants
4 Eigenvalues and Eigenvectors
5 Orthogonality
6 Partial Derivatives
7 Double and Triple Integrals
Fun topics Linear Algebra
Fun topics Multivariable Calculus

 

Exams

Midterm 1: Wednesday, March 5, 2-2:50 pm covers Lec 1-14

Practice Exam Solutions Comment
Study Guide
Mock Midterm 1 Solutions
Mock Midterm 2 Solutions
Mock Midterm 3 Solutions
Mock Midterm 4 Solutions
Mock Midterm 5 Solutions
Mock Midterm 6 Solutions Summer 2012 Midterm
Mock Midterm 7 Solutions Spring 2019 Midterm
Mock Midterm 8 Solutions Fall 2019 Midterm
Midterm 1 Solutions Spring 2025 Midterm
Midterm 1 – Review Solutions
Old Review Solutions
Old Quizzes Solutions
True/False Solutions Solution Video

Midterm 2: Wednesday, April 16, 2-2:50 pm, covers Lec 15-28 (LA refers to “Linear Algebra” and MC refers to “Multivariable Calculus”)

Practice Exam Solutions Topic Comments
Study Guide
Mock Midterm 1 Solutions LA
Mock Midterm 2 Solutions LA Summer 2012 Midterm
Mock Midterm 3 Solutions LA Winter 2019 Midterm
Mock Midterm 4 Solutions LA Winter 2019 Midterm
Mock Midterm 5 Solutions LA Fall 2019 Midterm
Mock Midterm 6 Solutions MC
Mock Midterm 7 Solutions MC
Mock Midterm 8 Solutions MC Spring 2018 Midterm
Mock Midterm 9 Solutions MC Spring 2018 Midterm
Mock Midterm 10 Solutions MC Spring 2021 Midterm
Mock Midterm 11 Solutions MC Fall 2021 Midterm
Midterm 2 Solutions Spring 2025 Midterm
Midterm 2 – Review
Old Review Solutions LA
Old Review Solutions MC
Old Quizzes Solutions LA
Old Quizzes Solutions MC
True/False Solutions LA Solution Video

Final Exam: Wednesday, May 14, 2-5 pm, covers the entire course

Calendar