Linear Algebra and Geometry 2

Much more about matrices; abstract vector spaces and their bases; linear transformations animated with MANIM

4.74 (183 reviews)
Udemy
platform
English
language
Math
category
3,604
students
47 hours
content
Mar 2024
last update
$79.99
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What you will learn

How to solve problems in linear algebra and geometry (illustrated with 153 solved problems) and why these methods work.

Important concepts concerning vector spaces, such as basis, dimension, coordinates, and subspaces.

Linear combinations, linear dependence and independence in various vector spaces, and how to interpret them geometrically in R^2 and R^3.

How to recalculate coordinates from one basis to another, both with help of transition matrices and by solving systems of equations.

Row space, columns space and nullspace for matrices, and about usage of these concepts for solving various types of problems.

Linear transformations: different ways of looking at them (as matrix transformations, as transformations preserving linear combinations).

How to compose linear transformations and how to compute their standard matrices in different bases; compute the kernel and the image for transformations.

Understand the connection between matrices and linear transformations, and see various concepts in accordance with this connection.

Work with various geometrical transformations in R^2 and R^3, be able to compute their matrices and explain how these transformations work.

Understand the concept of isometry and be able to give some examples, and formulate their connection with orthogonal matrices.

Transform any given basis for a subspace of R^n into an orthonormal basis of the same subspace with help of Gram-Schmidt Process.

Compute eigenvalues, eigenvectors, and eigenspaces for a given matrix, and give geometrical interpretations of these concepts.

Determine whether a given matrix is diagonalizable or not, and perform its diagonalization if it is.

Understand the relationship between diagonalizability and dimensions of eigenspaces for a matrix.

Use diagonalization for problem solving involving computing the powers of square matrices, and motivate why this method works.

Be able to formulate and use The Invertible Matrix Theorem and recognise the situations which are suitable for the determinant test (and which are not).

Use Wronskian to determine whether a set of smooth functions is linearly independent or not; be able to compute Vandermonde determinant.

Work with various vector spaces, for example with R^n, the space of all n-by-m matrices, the space of polynomials, the space of smooth functions.

Description

Linear Algebra and Geometry 2

Much more about matrices; abstract vector spaces and their bases


Chapter 1: Abstract vector spaces and related stuff


S1. Introduction to the course

S2. Real vector spaces and their subspaces

You will learn: the definition of vector spaces and the way of reasoning around the axioms; determine whether a subset of a vector space is a subspace or not.

S3. Linear combinations and linear independence

You will learn: the concept of linear combination and span, linearly dependent and independent sets; apply Gaussian elimination for determining whether a set is linearly independent; geometrical interpretation of linear dependence and linear independence.

S4. Coordinates, basis, and dimension

You will learn: about the concept of basis for a vector space, the coordinates w.r.t.\ a given basis, and the dimension of a vector space; you will learn how to apply the determinant test for determining whether a set of n vectors is a basis of R^n.

S5. Change of basis

You will learn: how to recalculate coordinates between bases by solving systems of linear equations, by using transition matrices, and by using Gaussian elimination; the geometry behind different coordinate systems.

S6. Row space, column space, and nullspace of a matrix

You will learn: concepts of row and column space, and the nullspace for a matrix; find bases for span of several vectors in R^n with different conditions for the basis.

S7. Rank, nullity, and four fundamental matrix spaces

You will learn: determine the rank and the nullity for a matrix; find orthogonal complement to a given subspace; four fundamental matrix spaces and the relationship between them.


Chapter 2: Linear transformations


S8. Matrix transformations from R^n to R^m

You will learn: about matrix transformations: understand the way of identifying linear transformations with matrices (produce the standard matrix for a given transformation, and produce the transformation for a given matrix); concepts: kernel, image and inverse operators; understand the link between them and nullspace, column space and inverse matrix.

S9. Geometry of matrix transformations on R^2 and R^3

You will learn: about transformations such as rotations, symmetries, projections and their matrices; you will learn how to illustrate the actions of linear transformations in the plane.

S10. Properties of matrix transformations

You will learn: what happens with subspaces and affine spaces (points, lines and planes) under linear transformations; what happens with the area and volume; composition of linear transformations as matrix multiplication.

S11. General linear transformations in different bases

You will learn: solving problems involving linear transformations between two vector spaces; work with linear transformations in different bases.


Chapter 3: Orthogonality


S12. Gram-Schmidt Process

You will learn: about orthonormal bases and their superiority above the other bases; about orthogonal projections on subspaces to R^n; produce orthonormal bases for given subspaces of R^n with help of Gram-Schmidt process.

S13. Orthogonal matrices

You will learn: definition and properties of orthonormal matrices; their geometrical interpretation.


Chapter 4: Intro to eigendecomposition of matrices


S14. Eigenvalues and eigenvectors

You will learn: compute eigenvalues and eigenvectors for square matrices with real entries; geometric interpretation of eigenvectors and eigenspaces.

S15. Diagonalization

You will learn: to determine whether a given matrix is diagonalizable or not; diagonalize matrices and apply the diagonalization for problem solving (the powers of matrices).

S16. Wrap-up Linear Algebra and Geometry 2

You will learn: about the content of the third course.


S17. Extras

You will learn: about all the courses we offer. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.


Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

A detailed description of the content of the course, with all the 214 videos and their titles, and with the texts of all the 153 problems solved during this course, is presented in the resource file

"001 List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_2.pdf”

under video 1 ("Introduction to the course"). This content is also presented in video 1.

Content

Introduction to the course

Introduction to the course

Real vector spaces and their subspaces

From abstract to concrete
From concrete to abstract
Our prototype
Formal definition of vector spaces Example 1: Rn
Vector spaces, Example 2: m × n matrices with real entries
Vector spaces, Example 3: real-valued functions on some interval
Vector spaces, Example 4: complex numbers
Cancellation property
Two properties of vector spaces; Definition of difference
Some properties of vector spaces
What is a subspace
All the subspaces in R2
All the subspaces in R3
Subspaces, Problem 1
Subspaces, Problem 2
Subspaces, Problem 3
Subspaces, Problem 4

Linear combinations and linear independence

Our unifying example
Linear combinations in Part 1
Linear combinations, new stuff. Example 1
Linear combinations Example 2
Linear combinations, Problem 1
Linear combinations, Problem 2
What is a span, definition and some examples
Span, Problem 3
Span, Problem 4
Span, Problem 5
What do we mean by trivial?
Linear independence and linear dependence
Geometry of linear independence and linear dependence
An important remark on linear independence in Rn
Linearly independent generators, Problem 6
Linear independence in the set of matrices, Problem 7
Linear independence in C^0[−∞, ∞], Problem 8
Vandermonde determinant and polynomials
Linear independence in C^∞(R), Problem 9
Wronskian and linear independence in C∞(R)
Linear independence in C^∞(R), Problem 10
Linear independence in C^∞(R), Problem 11

Coordinates, basis, and dimension

What is a basis and dimension?
Bases in the 3-space, Problem 1
Bases in the plane and in the 3-space
Bases in the 3-space, Problem 2
Bases in the 4-space, Problem 3
Bases in the 4-space, Problem 4
Bases in the space of polynomials, Problem 5
Coordinates with respect to a basis
Coordinates with respect to a basis are unique
Coordinates in our unifying example
Dimension of a subspace, Problem 6
Bases in a space of functions, Problem 7

Change of basis

Coordinates in different bases
It is easy to recalculate from the standard basis
Transition matrix, a derivation
Previous example with transition matrix
Our unifying example
One more simple example and bases
Two non-standard bases, Method 1
Two non-standard bases, Method 2
How to recalculate coordinates between two non-standard bases? An algorithm
Change of basis, Problem 1
Change of basis, Problem 2
Change of basis, Problem 3
Change of basis, Problem 4
Change of basis, Problem 5
Change to an orthonormal basis in R^2

Row space, column space, and nullspace of a matrix

What you are going to learn in this section
Row space and column space for a matrix
What are the elementary row operations doing to the row spaces?
What are the elementary row operations doing to the column spaces?
Column space, Problem 2
Determining a basis for a span, Problem 3
Determining a basis for a span consisting of a subset of given vectors, Prob
Determining a basis for a span consisting of a subset of given vectors, Prob
A tricky one: Let rows become columns, Problem 6
A basis in the space of polynomials, Problem 7
Nullspace for a matrix
How to find the nullspace, Problem 8
Nullspace, Problem 9
Nullspace, Problem 10

Rank, nullity, and four fundamental matrix spaces

Rank of a matrix
Nullity
Relationship between rank and nullity
Relationship between rank and nullity, Problem 1
Relationship between rank and nullity, Problem 2
Relationship between rank and nullity, Problem 3
Orthogonal complements, Problem 4
Four fundamental matrix spaces
The Fundamental Theorem of Linear Algebra and Gilbert Strang

Matrix transformations from R^n to R^m

What do we mean by linear?
Some terminology
How to think about functions from Rn to Rm?
When is a function from Rn to Rm linear? Approach 1
When is a function from Rn to Rm linear? Approach 2
When is a function from Rn to Rm linear? Approach 3
Approaches 2 and 3 are equivalent
Matrix transformations, Problem 1
Image, kernel, and inverse operators, Problem 2
Basis for the image, Problem 3
Kernel, Problem 4
Image and kernel, Problem 5
Inverse operators, Problem 6
Linear transformations, Problem 7
Kernel and geometry, Problem 8
Linear transformations, Problem 9

Geometry of matrix transformations on R^2 and R^3

Our unifying example: linear transformations and change of basis
An example with nontrivial kernel
Line symmetries in the plane
Projection on a given vector, Problem 1
Symmetry about the line y = kx, Problem 2
Rotation by 90 degrees about the origin
Rotation by the angle α about the origin
Expansion, compression, scaling, and shear
Plane symmetry in the 3-space, Problem 3
Projections on planes in the 3-space, Problem 4
Symmetry about a given plane, Problem 5
Projection on a given plane, Problem 6
Rotations in the 3-space, Problem 7

Properties of matrix transformations

What kind of properties we will discuss
What happens with vector subspaces and affine subspaces under linear transfo
Parallel lines transform into parallel lines, Problem 1
Transformations of straight lines, Problem 2
Change of area (volume) under linear operators in the plane (space)
Change of area under linear transformations, Problem 3
Compositions of linear transformations
How to obtain the standard matrix of a composition of linear transformations
Why does it work?
Compositions of linear transformations, Problem 4
Compositions of linear transformations, Problem 5

General linear transformations in different bases

Linear transformations between two linear spaces
Linear transformations, Problem 1
Linear transformations, Problem 2
Linear transformations, Problem 3
Linear transformations, Problem 4
Linear transformations, Problem 5
Linear transformations in different bases, Problem 6
Linear transformations in different bases
Linear transformations in different bases, Problem 7
Linear transformations in different bases, Problem 8
Linear transformations in different bases, Problem 9
Linear transformations, Problem 10
Linear transformations, Problem 11

Gram–Schmidt process

Dot product and orthogonality until now
Orthonormal bases are awesome
Orthonormal bases are awesome, Reason 1: distance
Orthonormal bases are awesome, Reason 2: dot product
Orthonormal bases are awesome, Reason 3: transition matrix
Orthonormal bases are awesome, Reason 4: coordinates
Coordinates in ON bases, Problem 1
Coordinates in orthogonal bases, Theorem and proof
Each orthogonal set is linearly independent, Proof
Coordinates in orthogonal bases, Problem 2
Orthonormal bases, Problem 3
Projection Theorem 1
Projection Theorem 2
Projection Formula, an illustration in the 3-space
Calculating projections, Problem 4
Calculating projections, Problem 5
Gram–Schmidt Process
Gram–Schmidt Process, Our unifying example
Gram–Schmidt Process, Problem 6
Gram–Schmidt Process, Problem 7

Orthogonal matrices

Product of a matrix and its transposed is symmetric
Definition and examples of orthogonal matrices
Geometry of 2-by-2 orthogonal matrices
A 3-by-3 example
Useful formulas for the coming proofs
Property 1: Determinant of each orthogonal matrix is 1 or −1
Property 2: Each orthogonal matrix A is invertible and A−1 is also orthogona
Property 3: Orthonormal columns and rows
Property 4: Orthogonal matrices are transition matrices between ON-bases
Property 5: Preserving distances and angles
Property 6: Product of orthogonal matrices is orthogonal
Orthogonal matrices, Problem 1
Orthogonal matrices, Problem 2

Eigenvalues and eigenvectors

Crash course in factoring polynomials
Eigenvalues and eigenvectors, the terms
Order of defining, order of computing
Eigenvalues and eigenvectors geometrically
Eigenvalues and eigenvectors, Problem 1
How to compute eigenvalues Characteristic polynomial
How to compute eigenvectors
Finding eigenvalues and eigenvectors: short and sweet
Eigenvalues and eigenvectors for examples from Video 180
Eigenvalues and eigenvectors, Problem 3
Eigenvalues and eigenvectors, Problem 4
Eigenvalues and eigenvectors, Problem 5
Eigenvalues and eigenvectors, Problem 6
Eigenvalues and eigenvectors, Problem 7

Diagonalization

Why you should love diagonal matrices
Similar matrices
Similarity of matrices is an equivalence relation (RST)
Shared properties of similar matrices
Diagonalizable matrices
How to diagonalize a matrix, a recipe
Diagonalize our favourite matrix
Eigenspaces; geometric and algebraic multiplicity of eigenvalues
Eigenspaces, Problem 2
Eigenvectors corresponding to different eigenvalues are linearly independent
A sufficient, but not necessary, condition for diagonalizability
Necessary and sufficient condition for diagonalizability
Diagonalizability, Problem 3
Diagonalizability, Problem 4
Diagonalizability, Problem 5
Diagonalizability, Problem 6
Diagonalizability, Problem 7
Powers of matrices
Powers of matrices, Problem 8
Diagonalization, Problem 9
Sneak peek into the next course; orthogonal diagonalization

Wrap-up Linear Algebra and Geometry 2

Linear Algebra and Geometry 2, Wrap-up
Yes, there will be Part 3!
Final words

Screenshots

Linear Algebra and Geometry 2 - Screenshot_01Linear Algebra and Geometry 2 - Screenshot_02Linear Algebra and Geometry 2 - Screenshot_03Linear Algebra and Geometry 2 - Screenshot_04

Reviews

Matthew
July 23, 2023
Good qualifications of the instructor. A lot of information is provided in the videos. This is the first time I understood the difference between linear transformations and matrix.
Cyrus
May 1, 2023
One of the best courses on linear algebra. I look forward to each video and it makes linear algebra learning fun.
Andrea
April 16, 2023
This course is extraordinary, like the first one as well. Topics are explained very well, from different angles and are always put in greater context. The many solved problems deepen the understandig and make you feel comfortable with the subjects. I absolutely recommend this course. It was a pleasure to watch and work through.
Alok
February 2, 2023
This has been an amazing course, making learning Linear Algebra a lot of fun! I would highly recommend it to everyone. Thank you!
Vamsikrishna
December 21, 2022
This course is the best on the internet—absolutely in-depth treatment of all topics pertaining to Linear Algebra. Prof. Wehlou has a knack for excitingly explaining things, and the lectures are a treasure mine when compared to other lectures (such as Prof. Strang, Prof. Pavel etc.). Waiting for all your other courses
Rajarshi
December 14, 2022
Gonna be honest it's probably the best way to learn linear algebra in an efficient and fast way if u give it effort and follow the problems (as in doing it before u watch the solution ) and the basic intuition and important abstractions are presented in a systematic way which will make math profs cry. Having all of the courses is a bonus as it flows really well very well. Is it worth the money ? honestly, yes id buys it at full price without even thinking about it.
Alfonso
November 21, 2022
Bien detallado el curso. Aprendí mucho e incluso algunos conceptos que antes no los tenía muy claros.
Justin
October 5, 2022
Much like the Linear Algebra 1 course, this course contains very thorough explanations of the information with lots of worked examples. I'm only to lecture 37, but up to this point, material from the first course is more clear (particularly the subspaces material, for me). I really enjoy her teaching.
Murat
September 13, 2022
Best Linear Algebra courses I took. Everything is clearly explained and proved in the lectures. Also, the instructor makes every effort to clear your doubts and help you. After completing these courses, most of the subjects in the machine learning seem very intuitive. I definitely recommend this series to anyone who wants to learn linear algebra.
Glenn
August 30, 2022
Very in-depth teaching of concepts with lots of exercises to solidify ones understanding of the teaching.
Hamish
July 27, 2022
I teach high school mathematics, sometimes touching on 1st year year university topics. This course is very thorough: I did about 75% of it as not all of it was relevant to my level. Hania is an excellent instructor and I look forward to doing other courses with her.
Jennifer
July 25, 2022
Thank you so much for this course. My lecturer at uni is new to the job and very hard to follow. The examples, diagrams and animations in this course really help me to understand the concepts!
Samir
May 26, 2022
Instructor is way more impressive than my expectation, she is kind helpful and informative. i wish you share this course to more and more geeks.
Jamieson
March 17, 2022
I struggled with linear algebra in college and have always felt sad I could not grasp the material. These lectures are a treat and a joy. Thank you so much for your contribution to mathematics and education.
Maciej
September 16, 2021
This course is very good, everything is very well explained and richly illustrated with pictures and computations.

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7/5/2021
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