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Pure Linear Algebra I

SAT Score Range

•

19 sessions

•

+17

This series was cancelled by the tutor on September 10, 2024. 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.

About

This is a full first course on proof-based linear algebra.
Sessions are 1h30m daily except for Sunday, so the course is at an accelerated pace.

Prerequisites:
-Algebra 2 with trig (with Calculus preferred but not required)

Syllabus:
(UNIT I) Basic Vector Operations
1. Vector geometry in R^n
2. Planes in R^3
3. Span, subspaces, and dimension
4. Basis and orthogonality
5. Cross Products

(UNIT II) Basic Vector Space Properties 2. Vector Spaces and Subspaces 3. Linear independence and Gram-Schmidt 4. Projections and Orthogonal Vector Decomposition

(Unit III) Linear Transformations and Motivating Matrices 5. Linear Transformations, Properties, and Kernel 6. Matrix operations, special matrices, and matrix properties

(Unit IV) Linear Systems 7. Determinants, Linear Systems, and Inverse Matrices
8. LU decomposition 9. Row, Column, and Null Space, Rank and Nullity

(Unit V) Representing Linear Transformations as Matrices 10. Change of Basis and Matrix Multiplication
11. Eigenvectors and Diagonalizability

(Unit VI) Special Topics in Applied Linear Algebra:
-Euclidean Inner Spaces
-Least squares regression
-Markov chains
-Raising numbers to matrices
-Solving differential equations using matrices

I will provide all reading and practice material.

Note: Required reading of 25 pages a week + Weekly homework (with feedback) + 1 test per unit (with review in-class and the tests graded with feedback).


Tutored by

Jose Roberto Cossich G 🇬🇹

Senior Tutor

Certified in 69 topics

View Profile

A little about me: I'm about the biggest Indie fan there is, I almost exclusively watch (John Hughes>>) 80s movies or the upcoming Star Wars series, and I read sci-fi in my spare time. I tutor for the sake of those aha moments, and have taken Multivariable Calc, Linear Algebra, Real Analysis, and Complex Analysis. Who knows what I'll tutor next, hope to see you soon.

Kindness C 🇨🇦

Certified in 17 topics

View Profile

Hi! I'm Kindness (aka Enci). As of 2024-25, I'm a junior year university student majoring in pure math! I graduated high school as part of the Class of 2023, and thanks to APs I skipped the freshman year of university. On Schoolhouse, I especially like to tutor in Enrichment as a study space host and advanced math tutor. I used to be quite active in tutoring the SAT, and I hope to return to doing that soon! In my free time (which I don't have much, lol), you can find me writing poems, playing the piano or violin, and scrapbooking.

Schedule

✋ ATTENDANCE POLICY

I try to make the session times as accommodating as possible, however make-up sessions are always available. Please feel free to ask for one if needed!

SESSION 1

10

Aug

SESSION 1

Other Topics

Other Topics

Sat 8:00 PM - 9:30 PM UTCAug 10, 8:00 PM - 9:30 PM UTC

Content Covered:
1. Vector geometry in R^n: -Vectors and Scalars
-Review of sigma notation
-Magnitude properties
-Vector law of sines and cosines, -Dot Product,

SESSION 2

12

Aug

SESSION 2

Other Topics

Other Topics

Mon 8:00 PM - 9:30 PM UTCAug 12, 8:00 PM - 9:30 PM UTC

-Parametric Equations, Lines, and Planes
-Finding the algebraic equation of planes using three points on the plane, the vector normal to the plane, and vice-versa
-The Cross Product
SESSION 3

13

Aug

SESSION 3

Other Topics

Other Topics

Tue 8:00 PM - 9:30 PM UTCAug 13, 8:00 PM - 9:30 PM UTC

-Review proof by contradiction
-Review proof by induction
-Introducing matrices, scalar multiplication and addition and proving their properties under addition
SESSION 4

14

Aug

SESSION 4

Other Topics

Other Topics

Wed 8:00 PM - 9:30 PM UTCAug 14, 8:00 PM - 9:30 PM UTC

Please have the homework for Unit I posted done before class.
Unit I Test (75m)

2. Vector Spaces and Subspaces
SESSION 5

16

Aug

SESSION 5

Other Topics

Other Topics

Fri 8:00 PM - 9:30 PM UTCAug 16, 8:00 PM - 9:30 PM UTC

Continuing discussion on vector spaces:
-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).
SESSION 6

17

Aug

SESSION 6

Other Topics

Other Topics

Sat 8:00 PM - 9:30 PM UTCAug 17, 8:00 PM - 9:30 PM UTC

Continuing our discussion on vector spaces.
SESSION 7

18

Aug

SESSION 7

Other Topics

Other Topics

Sun 8:00 PM - 9:30 PM UTCAug 18, 8:00 PM - 9:30 PM UTC

-Vector subspaces
(-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.)
SESSION 8

19

Aug

SESSION 8

Other Topics

Other Topics

Mon 8:15 PM - 9:45 PM UTCAug 19, 8:15 PM - 9:45 PM UTC

3. Linear independence and Gram-Schmidt
Content: -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)
SESSION 9

21

Aug

SESSION 9

Other Topics

Other Topics

Wed 8:00 PM - 9:30 PM UTCAug 21, 8:00 PM - 9:30 PM UTC

4. Projections and Orthogonal Decomposition
Content:
-Deriving and applying Fourier's Formula
-Projecting a point to a line
-Projecting a point to a plane
SESSION 10

23

Aug

SESSION 10

Other Topics

Other Topics

Fri 8:00 PM - 9:30 PM UTCAug 23, 8:00 PM - 9:30 PM UTC

Please have the homework posted done before class.
Unit II Homework Review and Test
SESSION 11

26

Aug

SESSION 11

Other Topics

Other Topics

Mon 8:00 PM - 9:00 PM UTCAug 26, 8:00 PM - 9:00 PM UTC

5. Linear Transformations, Properties, and Kernel
Topics covered:
- Defining linear transformations and finding matrix representations for them
- Properties of linear transformations
6. Matrix operations, special matrices, and matrix properties
Content covered:
-Rigorously defining matrix multiplication and properties of matrix multiplication
-Defining the transpose and its properties
-Row-Reduced Form
-Block, Triangular, Symmetric, and Skew-Symmetric Matrices

Please have the homework posted done before class.
Unit III Homework Review and Test
SESSION 12

27

Aug

SESSION 12

Other Topics

Other Topics

Tue 8:15 PM - 9:15 PM UTCAug 27, 8:15 PM - 9:15 PM UTC

7. Linear Systems, Detrminants, and Inverse Matrices
Content covered:
-Defining system consistency and the number of solutions to a system
-Exploring homogenous solutions
-Finding system solutions using Gaussian Elimination
-Calculating and interpreting the determinant for n=2,3 and n>3
-Determinant properties
SESSION 13

29

Aug

SESSION 13

Other Topics

Other Topics

Thu 9:45 PM - 10:45 PM UTCAug 29, 9:45 PM - 10:45 PM UTC

Please have the homework posted done before class.
Unit II Homework Review and Test
5. Linear Transformations, Properties, and Kernel
Topics covered:
- Defining linear transformations and finding matrix representations for them
- Properties of linear transformations
SESSION 14

30

Aug

SESSION 14

Other Topics

Other Topics

Fri 9:45 PM - 10:45 PM UTCAug 30, 9:45 PM - 10:45 PM UTC

6. Matrix operations, special matrices, and matrix properties
Content covered:
-Rigorously defining matrix multiplication and properties of matrix multiplication
-Defining the transpose and its properties
-Row-Reduced Form
-Block, Triangular, Symmetric, and Skew-Symmetric Matrices
SESSION 15

2

Sep

SESSION 15

Other Topics

Other Topics

Mon 9:45 PM - 10:45 PM UTCSep 2, 9:45 PM - 10:45 PM UTC

7. Linear Systems, Detrminants, and Inverse Matrices
Content covered:
-Defining system consistency and the number of solutions to a system
-Exploring homogenous solutions
-Finding system solutions using Gaussian Elimination
-Calculating and interpreting the determinant for n=2,3 and n>3
-Determinant properties
Please have the homework posted done before class.
Unit III Homework Review and Test
8. LU decomposition
Content covered:
-Finding matrix inverses using elementary row operations
-Properties of matrix inverses
-Using determinants to calculate inverses
SESSION 16

3

Sep

SESSION 16

Other Topics

Other Topics

Tue 9:45 PM - 10:45 PM UTCSep 3, 9:45 PM - 10:45 PM UTC

Please have the homework posted done before class.
Unit IV Homework Review and Test
SESSION 17

5

Sep

SESSION 17

Other Topics

Other Topics

Thu 1:45 AM - 2:45 AM UTCSep 5, 1:45 AM - 2:45 AM UTC

9. Row, Column, and Null Space, Rank and Nullity
Content covered:
-Defining the row and column space
-Determining the row rank and column rank
-The Rank-nullity theorem and non-trivial solutions to systems

SESSION 18

5

Sep

SESSION 18

Other Topics

Other Topics

Thu 9:45 PM - 10:45 PM UTCSep 5, 9:45 PM - 10:45 PM UTC

10. Change of Basis and Matrix Multiplication
-Introducing change of basis in order to motivate the definition of matrix multiplications
-Inverting change of basis matrices
SESSION 19

9

Sep

SESSION 19

Other Topics

Other Topics

Mon 9:45 PM - 10:45 PM UTCSep 9, 9:45 PM - 10:45 PM UTC

11. Eigeneverything and Diagonalizability
-Defining eigenvectors and eigenvalues
-Defining trace and the characteristic polynomial to find eigenvalues and eigenvectors
-Properties of eigenvalues
-Diagonalization of matrices using eigenvalues

Public Discussion

Please log in to see discussion on this series.

Aug 10 - Sep 9

5 weeks

60 - 90 mins

/ session