Math 121A – Linear Algebra

Welcome to Math 121A – Linear fun! In this course, we will look back at the beautiful topics from Math 3A and study them from a mathematician’s point of view. At the same time, we will strengthen our proof-writing skills, so that you’re prepared for the rest of your mathematics-adventure. This is fun for you and your whole family! Enjoy!!!

Syllabus

Here is the syllabus of the course: Syllabus

Lecture Notes and Videos

Lecture Notes YouTube Videos
Lecture 1 25 Examples of Vector Spaces
Linear Algebra Overview (optional)
Lecture 2 Subspace
Not subspace
Union of Subspaces
Lecture 3 Span is a subspace
Solving systems of equations
Lecture 4 Linear Independence
Intruder Theorem
Lecture 5 Dimension
Direct Sums
dim V + W
Lecture 6 Replacement Theorem
Replacement Theorem Proof (optional)
Infinite Dimensions
Lagrange Interpolation Formula (optional)
Every vector space has a basis (optional)
Lecture 7 Linear Transformations
Graph of a Linear Transformation
Nullspace is a subspace
Derivative and Linear Independence
Intersection of Range
Lecture 8 Rank-Nullity Theorem Proof
One-to-one and Onto
One to one iff Linearly Independent
Linear Extension Theorem
Linear Transformation with a given range
Lecture 9 Derivative in a box
Matrix of a matrix
Matrix with respect to a basis
Coordinates
Pseudo diagonalization
Lecture 10 Matrix Multiplication
Why Matrix Multiplication Works
AB vs. BA
Lecture 11 A^2 = A (optional)
A^2 = I (optional)
A^2 = O (optional)
Lecture 12 Left and Right Inverse (optional)
Matrix of T-1
Calculate T-1
Lecture 13 Isomorphism
Isomorphism and Dimension
2 Miracles of Linear Algebra
Differentiate with Linear Algebra
Lecture 14 Change of coordinates
Change of matrix
Reflection about a line
Projection on a line
Integrate with change of coordinates (optional)
Lecture 15 Midterm
Lecture 16 Dual spaces
Dual basis
How to find a dual basis
Lecture 17 Transpose Definition
Transpose Example
V**
Every basis is a dual basis
T^T one-to-one iff T onto
Annihilator of a subspace
Rank(A^T) = Rank(A)
Dual Lagrange Interpolation
V***** (optional)
Basis of a given dual basis (optional)
T^TT = T ? (optional)
Dirac Delta Functional (optional)
Lecture 18 Elementary Matrix
Unnecessary Operation?
Rank(A) = dim(Col(A))
Rank 1 Matrix (optional)
Lecture 19 Fundamental Rank Theorem
Rank(A^T) = Rank(A)
Product of elementary matrices
Find A^-1
Calculate T-1
Check A invertible
LU Decomposition (optional)
Lecture 20 Gaussian Elimination
Row operations preserve solutions
Every Matrix has a REF
Uniqueness of RREF (optional)
Lecture 21 Nul(A), Col(A), Rank(A)
Rank and Existence of Solutions
Rank Criterion Consequences
Lecture 22 Homogeneous and Particular Solution
Find a basis
Basis Extension
Reconstructing Matrix from RREF
Lecture 23 Area of a Parallelogram
ad – bc = 0
Determinants and Bomberman
Lecture 24 Multilinearity of Determinant
Determinant Puzzle
Expand along any row
Determinants and Row-Reduction
Determinants and Row-Reduction Example
Lecture 25 Det(AB) = Det(A) Det(B)
Det(A^-1) = 1/Det(A)
Det(A^T) = Det(A)
Vandermonde Determinant (Classic way)
Vandermonde Determinant (OMG Way) (optional)
Neat Determinant
Det of Block Matrix
Cramer’s Rule
Cramer’s Rule Proof
Determinants and Inverses (optional)
Adjugate Matrix (optional)
Determinants and Volumes (optional)
Characterization of det (optional)
Lecture 26 Eigenvalues and Eigenvectors
Abstract Eigenvalues and Eigenvectors
Oh Shift !
Lecture 27 Diagonalization and Legend of Zelda
Characterizations of Diagonalizability
Eigenvectors and Geometry
Lecture 28 Eigenvectors and Linear Independence
Not Diagonalizable
Lecture 29 Ultimate Diagonalization Test
Ultimate Diagonalization Test 2
Derivative not diagonalizable
Diagonalization and Direct Sums
Square root of a matrix (optional)
Matrix Limits and Markov Chains (optional)
111 True/False Questions (optional)

 

Homework

Homework will be posted here. Homework is due on Wednesdays during discussion. The lowest two homework assignments are dropped.

Note: The solutions don’t correspond exactly to the problems I’ve assigned, but there’s significant overlap. They are written by Janak Ramakrishnan, a TA for this course back in 2007 when I took it at UC Berkeley. If there’s a specific problem you’d like to see the solutions for, please let me or the TA know and we’d be happy to write it up for you.

Solutions: Here’s a link to solutions to all the problems in the book: Solutions

Homework Due Date Related Solutions
Homework 1 Wednesday, April 3 Solutions + AP Solutions
Homework 2 Wednesday, April 10 Solutions
Homework 3 Wednesday, April 17 Solutions + AP Solutions
Homework 4 Wednesday, April 24 Solutions + AP Solutions
Homework 5 Wednesday, May 1 Solutions
Homework 6 Wednesday, May 8 Solutions + AP4 Solution
Homework 7 Wednesday, May 15 Solutions + AP Solutions
Homework 8 Wednesday, May 22 Solutions + AP Solutions
Homework 9 Wednesday, May 29 Solutions + AP Solutions
Homework 10 Wednesday, June 5 Solutions + AP Solutions
Non Homework Never due Solutions

 

Useful links

Canvas Course Site (to check your quiz and exam scores)

My Youtube Channel

Linear Algebra – Playlist

Exams

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

Midterm Exam: Friday, May 3, 11 – 11:50 AM in 140 PSCB

Covers Chapters 1 and 2 (up to and including 2.4)

Study Guide: Study Guide

Practice Exam: Mock Midterm (Solutions)

Midterm Review: Midterm Review

Chapter 1 – Playlist: Vector Spaces

Chapter 2 – Playlist: Linear Transformations and Matrices

Midterm Exam: Midterm (Solutions)

 

Final Exam: Tuesday, June 11, 1:30 – 3:30 PM in 140 PSCB

Covers Chapters 1-5 (up to and including 5.2)

Extra OH: F 2-3, M 3-5

Study Guide: Study Guide

Practice Exam: Mock Final (Solutions)

Final Review: Final Review

Final Exam: Final Exam (Solutions)

Solutions to all the problems in the book: Solutions

YouTube Playlists:

Playlist Link
Chapter 1 Vector Spaces
Chapter 2 Linear Transformations and Matrices
Section 2.6 Dual Spaces
Chapter 3 Linear Equations
Chapter 4 Determinants
Chapter 5 Diagonalization
111 T/F Questions 111 T/F Questions