MATH 3A

Welcome to Math 3A – linear fun! In this course, we will learn how to solve systems of equations and interpret them geometrically, and there will be lots of miracles. This is math at its finest! On this page you can find the syllabus, suggested homework, and practice exams. Enjoy the ride!

Syllabus

Math 3A – Syllabus

Lecture Notes and Videos

#	Lecture Notes	YouTube Videos
1	Systems of Equations	Row-reduction
		Inconsistent Systems
2	Row-Reduction and Echelon Forms	Infinitely many solutions
		Solving ax + by
3	Vectors and Ax = b
4	Span and Ax = 0	Span
		Row Theorem
5	Linear Independence	Linear Independence
6	Linear Transformations (I)	Showing T is linear
7	Linear Transformations (II)	Matrix of T and One-to-one and Onto
		Rotation Matrix
8	Matrix Algebra	Matrix Multiplication
		AB vs BA
9	The Inverse of a Matrix	Finding A-1
		Product of elementary matrices (optional)
		LU Factorization (optional)
10	Invertible Matrix Theorem + 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
11	Nul(A) and Col(A)	Nul(A)
		Nul(A) is a subspace
		Checking something is a basis
12	Dimension and Rank	Basis and dimension
		Nul(A), Col(A), Rank(A)
		Coordinates
		Infinite-dimensional space (optional)
		Rank 1 Matrices and Outer Products (optional)
13	Determinants	Determinants
		Determinants and row-reduction
		More determinants and row-reduction
		Determinants and Invertibility
14	Applications of Determinants	Cramer’s Rule
		A-1 using determinants
		Area of parallelogram
		Determinants and volumes
15	Midterm – Review	111 True/False Questions
16	Midterm
17	Eigenvalues and Eigenvectors	Diagonalize 2×2 matrix
18	Diagonalization	Diagonalize 3×3 matrix
		Diagonalization analogy
		Complex Eigenvalues (optional)
19	Diagonalizability	Not diagonalizable
20	B-Matrices	B-matrix of A
		Derivative in a box (optional)
		Matrix of a matrix (optional)
21	Applications of Diagonalization	Pokemon Battle
		Fibonacci Sequence
		Matrix Limit 1
		Matrix Limit 2
		Square root of a matrix
		Matrix exponential
		Cos of a matrix
		Systems of differential equations (optional)
22	Dot products and lengths
23	Orthogonality Awesomeness	Orthogonal Sets
24	Orthogonal Projections	Orthogonal Projections
		Proof of Cauchy-Schwarz
		Awesome Cauchy-Schwarz (optional)
25	The Gram-Schmidt Process	Gram-Schmidt Process
		QR Decomposition
26	Least-Squares	Least Squares
		Linear Models
27	Final Review 1	Symmetric Matrices (optional)
		Quadratic Forms (optional)
28	Final Review 2
29	Final Review 3	Linear Algebra Overview
		111 True/False Questions

Suggested Homework

Suggested Homework is not to be turned in, but important for the quizzes and the exams. Remember that the lowest 2 quizzes are dropped and that there are no quizzes on Th 09/26 (Week 0) and on Th 11/28 (Thanksgiving)

Hints/Solutions to some of the problems: Solution – Bank (from an older edition, so be careful)

Homework	“Due Date”	Problems
Homework 1	October 3	1.1: 11, 13, 21, 23, 24, 28 1.2: 11, 19, 21, 22, 24, 25, 30 1.3: 11, 17, 23, 24, 25, 26 1.4: 23, 24, 25, 31
Homework 2	October 10	1.4: 22 1.5: 9, 11, 23, 24, 31, 32 1.7: 2, 7, 11, 17, 21, 22, 27, 34, 37 1.8: 15, 17, 21, 32, 33, 35 1.9: 3, 8, 10
Homework 3	October 17	1.8: 3, 11, 22, 34 1.9: 23, 24, 25, 27 2.1: 2, 3, 9, 10, 15, 16, 24, 25, 27 2.2: 1, 5, 9, 10, 18, 21, 29, 31
Homework 4	October 24	2.3: 5, 11, 12, 16, 18, 30, 33 2.8: 1, 2, 7, 9, 13, 17, 21, 22, 25, 31 2.9: 1, 3, 10, 16, 17, 18, 20, 25
Homework 5	October 31	3.1: 3, 13, 37, 39, 40 3.2: 5, 23, 27, 28, 32, 33, 34, 35, 39 3.3: 3, 13, 18, 22, 31, 32
Homework 6	November 7	5.1: 5, 10, 15, 21, 22, 24, 25, 26, 32 5.2: 1, 7, 9, 15, 19, 21, 22, 24
Homework 7	November 14	5.3: 1, 9, 13, 21, 22, 23, 24, 27, 31, 32 5.4: 11, 13, 17, 19, 20, 21, 22, 23
Homework 8	November 21	5.3: Applications 6.1: 1, 5, 10, 14, 17, 19abc, 20abd, 22, 24, 28 6.2: 1, 7, 17, 20, 23abc, 24ab
Homework 9	December 5	6.1: 19de, 20ce, 31 6.2: 11, 13, 15, 23de, 24cde, 25 6.3: 3, 10, 11, 17, 21, 22 6.4: 3, 7, 9, 17, 18 6.5: 3, 7, 9, 13, 15, 17, 18, 24, 25 6.6: 4

Useful links

My Youtube Channel

Playlist	Link
Chapter 1	Linear Equations in Linear Algebra
Chapter 2	Matrix Algebra
Chapter 3	Determinants
Chapter 5	Eigenvalues and Eigenvectors
Chapter 6	Orthogonality
Fun topics	Linear Algebra

Exams

Here you can find information about the exams, as well as other goodies such as practice exams.

Midterm Exam: Friday, November 1, 2 – 2:50 PM in 178 HH

Covers Chapters 1, 2, and 3

Study Guide	Study Guide
Chapter 1 – Playlist	Linear Equations
Chapter 2 – Playlist	Matrix Algebra
Chapter 3 – Playlist	Determinants

Practice Exam	Solutions	Comments
Peyam 1	Solutions	Midterm (Fall 2019)
Peyam 2	Solutions	Midterm (Winter 2019)
Peyam 3	Solutions
Peyam 4	Solutions
Peyam 5	Solutions
Vojta 1	Solutions	Ignore 1ef and 5
Vojta 2	Solutions
Ribet	Solutions	Ignore 1, 2abe, 3b
Sample T/F	Solutions	Ignore 1g, 2cef, 3i, 4fgh
Old Review	Solutions	Ignore 5, 10, 11
Old Quiz 1	Solutions
Old Quiz 2	Solutions
Old Quiz 3	Solutions
Old Quiz 4	Solutions	Ignore 1
Old Quiz 5	Solutions	Only do 1bcd
111 True-False	Solutions Video	Only do up to 70, and ignore 19, 41, 42, 52, 54, 59, 66, 68

Final Exam: Friday, December 13, 1:30 – 3:30 pm in 178 HH

Covers Chapters 1, 2, 3, 5, and 6

Warning: The Final Exam is cumulative, so make sure to also look at all the practice midterms for a complete review. Also, ignore any questions about row-spaces

Study Guide	Study Guide
Chapter 1 – Playlist	Linear Equations
Chapter 2 – Playlist	Matrix Algebra
Chapter 3 – Playlist	Determinants
Chapter 5 – Playlist	Eigenvalues and Eigenvectors
Chapter 6 – Playlist	Orthogonality
111 True-False Questions	111 True-False Questions

Practice Exam	Solutions	Comments
Peyam 1	Solutions	Final Exam (Fall 2019)
Peyam 2	Solutions	Final Exam (Winter 2019)
Peyam 3	Solutions	Final Exam (Winter 2019)
Peyam 4	Solutions	Only do 1-3
Peyam 5	Solutions	Only do 1-3
Peyam 6	Solutions	Only do 1abcd, 2, 3, 6
Peyam 7	Solutions	Only do 1abde, 2, 4, 8, 9
Peyam 8	Solutions	Ignore 1bf, 6, 7, 8, 10
Canez	Solutions	Only do 1, 4, 6-10
Vojta	Solutions	Ignore 1a, 7
Tataru	Solutions	Ignore 4, 5bd
Ribet	Solutions	Ignore 1a
Bergman	Solutions	Ignore 2ce, 5
Sample T/F	Solutions	Ignore 3a, 4bcdi
111 True-False	Solutions YouTube Video	Ignore 19, 41, 42, 52, 54, 59, 66, 68, 85, 86, 104, 108 – 111
Old Review	Solutions	Only do 1, 3, 5
Old Quiz 6	Solutions	Only do 1
Old Quiz 7	Solutions