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Calculus • Series

# A First (Rigorous) Course in Linear Algebra

Jose Roberto Cossich G

This series was cancelled by the tutor on October 7, 2023. We're very sorry–you can explore more Calculus series here. All 1:1 and group chats related to this series are disabled 7 days after the last session.

### Series Details

About

Hi, everyone! I'm proud to present Schoolhouse's first series on Linear Algebra! In this class we will cover all the standard topics of a first class in Linear Algebra (typically take sometime after Calc 2 for greater mathematical maturity, though it can be taken after Calc 3 or before Calculus in the first place.) To be clear, here are the pre-requisites: Algebra (1 and 2, and including Trig) are mandatory. Calculus 1 and 2 are strongly recommended just for maturity, but not required (if you get straight A's in all of Algebra 2 and Trig you should mostly be fine); Calc 3 helps as well (and when I set up my Calc 3 series we will be employing Linear Algebra so if that interests you this course is recommended.) I've taken Multivariable Calculus with Linear Algebra (applied) as well as Linear Algebra itself (proof-based) at Stanford OHS and Stanford ULO, getting an average of B+ on both. Generally, here are the topics we will cover (each week): 1. Basic vector operations in R^n 2. Vector Spaces and Subspaces induction technique. 3. Linear independence and Gram-Schmidt 4. Projections and Fourier's Formula 5. Linear Transformations, Properties, and Kernel 6. Matrix operations, special matrices, and matrix properties. 7. Determinants, Linear Systems, Inverse Matrices, and LU decomposition 8. Row, Column, and Null Space, Rank and Nullity 9. Change of Basis, Eigeneverything, and Diagonalizability 10. TBD: QR Decomposition and more on Transpose, Orthogonal Complements, Markov Chains, or anything else. The course expectation is at least 5 hours per week apart from class, office hours, and quizzes. (About 3 hours reading and 2 hours doing homework.) We will meet every Saturday at the same time. In class we will not cover all the content (there is simply enough time), rather we will discuss and go over the proofs the main results, which also appear in the text. We will skip over proofs and results whose proofs we all understand. We will focus on examples and practice in preparation for the assigned exercises and quizzes. We will use Bronson's "Linear Algebra - Algorithms, Applications, and Techniques" as our main text, ( http://ndl.ethernet.edu.et/bitstream/123456789/24629/1/Richard%20Bronson.pdf ) in addition to Stanford's Math 51 text (fragments taken under fair use law which will be posted gradually).

✋ ATTENDANCE POLICY

Please do not miss class. We move very fast, and you need to organize your time in order to complete all the work.

If you do, you are free to attend office hours, request catch-up classes, and the independent material (reading, homework, quizzes, etc.)

Dates

September 1 - September 30

Learners

9 / 12

Total Sessions

6

### About the Tutor

Hey, my name's Jose, and I'm currently a senior. I'm about the biggest Indie music fan there is, I love 80s movies (John Hughes goat), and reading in my spare time. I enjoy math because it allows me to express myself artistically, and tutor for the mere sake of those aha moments. I've gotten 5s on AP Spanish Lit and AP English Lang, and I've taken Multivariable Calc and Linear Algebra (which I hope to tutor soon!) I'm currently taking AP Physics C and Real Analysis.

View Jose Roberto Cossich G's Profile

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### Past Sessions

6
1
Sep

Session 1

#### Other Topics

*WEEK 1* (Class time: 1h15m) READING: None required for the first session. CONTENT: -Orientation -Vectors and Scalars -Parametric Equations, Lines, and Planes, -Dot Product, -Vector law of sines and cosines, The Generalized Pythagorean Theorem and the Cauchy–Schwarz Inequality, -Orthogonality and Orthonormality, The Normal Vector and Cross Product
2
Sep

Session 2

#### Other Topics

*WEEK 1* (Class time: 45m) READING: Chapters 1-3 from Math 51 (fragments included in the assessments page), derivation of the cross product (pdf included), CONTENT: We will finish covering last class's content.
14
Sep

Session 3

#### Other Topics

*WEEK 2* (Class time: 1h 15m) READING: We will be using the exercises in the first chapter of Ross's "Elementary Analysis" for induction (pages 4-6 (https://honorsanalysis.math.gatech.edu/sites/default/files/documents/Ross_analysis.pdf ); Sections 2.2 of Bronson and 1.3 (pg.13-19) of Ross (linked in previous session description. CONTENT: -Induction techniques and summation notation -Rigorous definition of a vector, vector space, as well as their properties (the 10 requirements of spaces, scalar multiplication by zero, scalar multiplication of the zero vector, uniqueness of the additive inverse and the zero vector, the additive inverse by scalar multiplication and the existence of a two-sided or one-sided inverse.
16
Sep

Session 4

#### Other Topics

*WEEK 3* READING: Section 2.3-2.5 of Bronson, and the fragments from Math 51 (pages 392 396-398) in the assessments page. CONTENT: -Definition of vector subspace and their 2 conditions. Developing the subspace containing the zero vector, the idea of an n-dimensional planes including the origin as a subspace, the spanning set, span as a subspace, and definition of a field. -Definition of linear (-in)dependency through linear combinations -The subspace created by a linearly independent and dependent set -Definition of basis, dimension, and span as a subspace. -The superset of a linearly dependent and independent set -Uniqueness of a basis representation -Every basis of the same subspace has the same dimension (number of vectors)
23
Sep

Session 5

#### Other Topics

*WEEK 4* -Continue covering last session READING: From Math 51, Chapter 5 page 91. Chapter 6 of Math 51 (fragments attached in the assessments page) CONTENT: -Definition of projection and the derivation of the projection formula. -Derivation of Fourier's Formula (more commonly known as The Orthogonal Decomposition Theorem) and visualization.
30
Sep

Session 6

#### Other Topics

*WEEK 5* READING: Sections 3.1-3.3, 3.5 (except for pages 386-388) from Bronson CONTENT: -Definition of a relation, function, domain, codomain, range and image -How to read and make mapping diagrams -Definition of a linear transformation -Examples of linear and non-linear transformations (and how to distinguish them) -Developing the idea of a matrix and that of matrix multiplication from them -Coordinate representation of a vector through basis vectors.

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