Linear Algebra and Geometry 1

Systems of equations, matrices, determinants, vectors, and geometry of straight lines and planes in the 3-space

4.95 (742 reviews)
Udemy
platform
English
language
Math
category
6,110
students
47 hours
content
Mar 2024
last update
$84.99
regular price

What you will learn

How to solve problems in linear algebra and geometry (illustrated with 175 solved problems) and why these methods work.

Solve systems of linear equations with help of Gauss-Jordan or Gaussian elimination, the latter followed by back-substitution.

Interpret geometrically solution sets of systems of linear equations by analysing their RREF matrix (row equivalent with the augmented matrix of the system).

Matrix operations (addition, scaling, multiplication), how they are defined, how they are applied, and what computational rules hold for them.

Matrix inverse: determine whether a matrix is invertible; compute its inverse: both with (Jacobi) algorithm and by the explicit formula; matrix equations.

Determinants, their definition, properties, and different ways of computing them; determinant equations; Cramer's rule for n-by-n systems of equations.

Vectors, their coordinates and norm; geometrical vectors and abstract vectors, their addition and scaling: arithmetically and geometrically (in 2D and 3D).

Vector products (scaling, dot product, cross product, scalar triple product), their properties and applications; orthogonal projection and vector decomposition.

Analytical geometry in the 3-space: different ways of describing lines and planes, with applications in problem solving.

Compute distances between points, planes and lines in the 3-space, both by using orthogonal projections and by geometrical reasoning.

Determine whether lines and planes are parallel, and compute the angles between them (using dot product and directional or normal vectors) if they intersect.

How to geometrically interpret n-by-2 and n-by-3 systems of equations and their solution sets as intersection sets between lines in 2D or planes in 3D.

Understand the connection between systems of linear equations and matrix multiplication.

Invertible Matrix Theorem and its applications; apply determinant test in various situations.

Description

Linear Algebra and Geometry 1

Systems of equations, matrices, vectors, and geometry


Chapter 1: Systems of linear equations


S1. Introduction to the course

S2. Some basic concepts

You will learn: some basic concepts that will be used in this course. Most of them are known from high-school courses in mathematics, some of them are new; the latter will appear later in the course and will be treated more in depth then.

S3. Systems of linear equations; building up your geometrical intuition

You will learn: some basic concepts about linear equations and systems of linear equations; geometry behind systems of linear equations.

S4. Solving systems of linear equations; Gaussian elimination

You will learn: solve systems of linear equations using Gaussian elimination (and back-substitution) and Gauss--Jordan elimination in cases of systems with unique solutions, inconsistent systems, and systems with infinitely many solutions (parameter solutions).

S5. Some applications in mathematics and natural sciences

You will learn: how systems of linear equations are used in other branches of mathematics and in natural sciences.


Chapter 2: Matrices and determinants


S6. Matrices and matrix operations

You will learn: the definition of matrices and their arithmetic operations (matrix addition, matrix subtraction, scalar multiplication, matrix multiplication). Different kinds of matrices (square matrices, triangular matrices, diagonal matrices, zero matrices, identity matrix).

S7. Inverses; Algebraic properties of matrices

You will learn: use matrix algebra; the definition of the inverse of a matrix.

S8. Elementary matrices and a method for finding A inverse

You will learn: how to compute the inverse of a matrix with Gauss-Jordan elimination (Jacobi’s method).

S9. Linear systems and matrices

You will learn: about the link between systems of linear equations and matrix multiplication.

S10. Determinants

You will learn: the definition of the determinant; apply the laws of determinant arithmetics, particularly the multiplicative property and the expansion along a row or a column; solving equations involving determinants; the explicite formula for solving of n-by-n systems of linear equations (Cramer's rule), the explicite formula for inverse to a non-singular matrix.


Chapter 3: Vectors and their products


S11. Vectors in 2-space, 3-space, and n-space

You will learn: apply and graphically illustrate the arithmetic operations for vectors in the plane; apply the arithmetic operations for vectors in R^n.

S12. Distance and norm in R^n

You will learn: compute the distance between points in R^n and norms of vectors in R^n, normalize vectors.

S13. Dot product, orthogonality, and orthogonal projections

You will learn: definition of dot product and the way you can use it for computing angles between geometrical vectors.

S14. Cross product, parallelograms and parallelepipeds

You will learn: definition of cross product and interpretation of 3-by-3 determinants as the volume of a parallelepiped in the 3-space.


Chapter 4: Analytical geometry of lines and planes


S15. Lines in R^2

You will learn: several ways of describing lines in the plane (slope-intercept equation, intercept form, point-vector equation, parametric equation) and how to compute other kinds of equations given one of the equations named above.

S16. Planes in R^3

You will learn: several ways of describing planes in the 3-spaces (normal equation, intercept form, parametric equation) and how to compute other kinds of equations given one of the equations named above.

S17. Lines in R^3

You will learn: several ways of describing lines in the 3-space (point-vector equation, parametric equation, standard equation) and how to compute other kinds of equations given one of the equations named above.

S18. Geometry of linear systems; incidence between lines and planes

You will learn: determine the equations for a line and a plane and how to use these for computing intersections by solving systems of equations.

S19. Distance between points, lines, and planes

You will learn: determine the equations for a line and a plane and how to use these for computing distances.

S20. Some words about the next course

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


S21. 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 222 videos and their titles, and with the texts of all the 175 problems solved during this course, is presented in the resource file

“001 Outline_Linear_Algebra_and_Geometry_1.pdf”

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

Content

Introduction to the course

Introduction

Some basic concepts

Coordinate systems and coordinates in the plane and in the 3-space.
Slope-intercept equations of straight lines in the plane
Normal equations of planes in the 3-space.
Vectors
Scalars
Vector addition and vector scaling
Linear combinations
Matrices
Linear transformations
Matrix—vector multiplication
Rules for computations with real numbers
Pythagorean Theorem and distance between points
Sine, cosine, and pythagorean identity
Cosine Rule

Systems of linear equations; building up your geometrical intuition

Different ways of looking at equations
Solution set
Linear and non-linear equations
Systems of linear equations
Solution sets of systems of linear equations
An example of a 2 × 2 system of linear equations, a graphical solution
Possible solution sets of 2 × 2 systems of linear equations
Possible solution sets of 3 × 2 systems of linear equations
Possible solution sets of 3 × 3 systems of linear equations
Possible solution sets of 2 × 3 systems of linear equations
Possible solution sets of m × n systems of linear equations

Solving systems of linear equations; Gaussian elimination

Our earlier problem revisited; an algebraical solution
Three elementary operations
What is Gauss—Jordan elimination and Gaussian elimination?
Gauss—Jordan elimination, a 2-by-2 system with unique solution
The same example solved with Gaussian elimination and back-substitution
The same example solved with matrix operations; coefficient matrix and augmented
How to write the augmented matrix for a given system of equations, Problem 1
How to write system of equations to a given augmented matrix, Problem 2
Gaussian elimination, Problem 3
Gaussian elimination, Problem 4
Gaussian elimination, Problem 5
Gaussian elimination, Problem 6.
What happens if the system is inconsistent?
Gaussian elimination, Problem 7.
Preparation to the general formulation of the algorithm; REF and RREF matrices.
How to read solutions from REF and RREF matrices?
General formulation of the algorithm in Gauss–Jordan elimination
Gauss–Jordan elimination, Problem 8
Gauss–Jordan elimination, Problem 9
Gaussian elimination, Problem 10
Gauss–Jordan elimination, Problem 11
Gauss–Jordan elimination, Problem 12
Gauss–Jordan elimination, Problem 13

Some applications in mathematics and natural sciences

Solving systems of linear equations in Linear Algebra and Geometry
Solving systems of linear equations (Calculus) Problem 1
Solving systems of linear equations (Calculus) Problem 2
Solving systems of linear equations (Calculus) Problem 3
Solving systems of linear equations (Calculus) Problem 4
Problem 5 (Chemistry)
Problem 6 (Electrical circuits)

Matrices and matrix operations

Introduction to matrices
Different types of matrices
Matrix addition and subtraction, Problem 1
Matrix scaling, with geometrical interpretation
Matrix scaling, Problem 2
Matrix multiplication, with geometrical interpretation
Matrix multiplication, how to do
Matrix multiplication, Problem 3
Matrix multiplication and systems of equations, Problem 4
Transposed matrix, definition and some examples
Trace of a matrix, definition and an example
Various matrix operations, Problem 7
Various matrix operations, Problem 8

Inverses; Algebraic properties of matrices

Properties of matrix operations, an introduction
Matrix addition has all the good properties
Matrix multiplication has a neutral element for square matrices
Matrix multiplication is associative
Matrix multiplication is not commutative
Sometimes commutativity happens, Problem 1
Two distributive laws
Matrix multiplication does not have the zero-product property
There is no cancellation law for matrix multiplication
Inverse matrices; not all non-zero square matrices have an inverse
Inverse matrix for 2-by-2 matrices; non-zero determinant
Solving matrix equations, Problem 2
Powers of matrices; powers of diagonal matrices
Computation rules for transposed matrices
Supplement to Video 83; Inverse of a product
Inverse of a transposed matrix
Various rules, Problem 3

Elementary matrices and a method for finding A inverse

Inverse matrices, introduction to the algorithm
Algorithm for inverse matrices, an example
Matrix inverse, Problem 1
Matrix inverse, Problem 2
Matrix equations, Problem 3
Matrix equations, Problem 4
Matrix equations, Problem 5
Matrix equations, Problem 6
Matrix inverse, Problem 7
Elementary operations and elementary matrices
Inverse elementary operations and their matrices
A really important theorem
Four equivalent statements

Linear systems and matrices

Formally about the number of solutions to systems of linear equations
Two more statements in our important theorem
Solution of a linear system using A inverse, Problem 1
Determining consistency by elimination, Problem 2
Matrix equations, Problem 3

Determinants

Why the determinants are important
2-by-2 determinants; notation for n-by-n determinants
Geometrical interpretations of determinants
Geometrically about the determinant of a product
Definition of determinants
Conclusion 1: Determinant of matrices with interchanged columns
Conclusion 2: What happens when one column is a linear combination of others
Conclusion 3: About adding a multiple of a column to another column
Conclusion 4: Determinant of kA for any k ∈ R
Elementary column operations
How to compute 2-by-2 determinants from the definition
How to compute 3-by-3 determinants from the definition
Sarrus’ rule for 3-by-3 determinants
Determinant of transposed matrix; row operations
Evaluating determinants by cofactor expansion along rows or columns
Evaluating determinants by row or column reduction
Determinant of inverse
Properties of determinants, Problem 1
Properties of determinants, Problem 2
Properties of determinants, Problem 3
Determinant equations, Problem 4
Determinant equations, Problem 5
Determinant equations, Problem 6
Determinant equations, Problem 7
Invertible matrices, determinant test with a proof, Problem 8
Cramer’s rule, a proof, an example, and a geometrical interpretation
Cramer’s rule, Problem 9
Inverse matrix, an explicit formula
Invertible matrices, Problem 10
Problem 11, a large determinant
Problem 12, another large determinant
Problem 13: a trigonometric determinant
Problem 14: Vandermonde determinant

Vectors in 2-space, 3-space, and n-space

Vectors, a repetition
Computation rules for vector addition and scaling
Computations with vectors, Problem 1
Computations with vectors, Problem 2
Computations with vectors, Problem 3
Parallel vectors, Problem 4
Parallel vectors, Problem 5
Linear combinations, Problem 6
Linear combinations, Problem 7
Linear combinations, linear independence, Problem 8
Linear combinations, linear dependence, Problem 9
Area, Problem 10
Midpoint of a line segment, Problem 11

Distance and norm in R^n

Norm of a vector, Problem 1
Properties of the norm
Distance between points, Problem 2
Unit vectors, how to normalize a vector
Unit vectors in given direction, Problem 3

Dot product, orthogonality, and orthogonal projections

Different products for vectors
Perpendicular straight lines and orthogonal vectors
Orthogonal projections
Definition of dot product for geometrical vectors
How to compute dot product, an example
Dot product for vectors in R^n; orthogonality and angles
Properties of dot product
Angles between vectors, Problem 1
Angles between vectors, Problem 2
How to find vector orthogonal to a given vector in the plane or in the 3-space
Orthogonal projections and decompositions
Orthogonal projections and decompositions, Problem 3
Orthogonal projections and decompositions, Problem 4
Orthogonal sets, Problem 5
Orthogonal projections and decompositions, Problem 6

Cross product, parallelograms and parallelepipeds

Cross product, an introduction
Cross product, how it is defined
Three properties of cross product
The length of the cross product of two vectors
More properties of cross product
Cross product: Problem 1
Cross product in the plane
Scalar triple product and volume
Scalar triple product, Problem 2
Collinearity in the plane and coplanarity in the 3-space
Determinant test for vectors, Problem 3

Lines in R^2

Lines in the plane, an introduction
Slope-intercept and intercept form
Normal equation
Parametric equations
Determinant equation
Lines in the plane, Problem 5

Planes in R^3

Planes in the 3-space, an introduction
Normal and intercept equation
Parametric equations
Parametric to normal
Normal to parametric; find a point and two parallel vectors for a given plane
Determinant equation
Planes: Problem 6

Lines in R^3

Lines in the 3 space, an introduction
Lines in the 3 space, Problem 1
Lines in the 3 space, Problem 2
Lines in the 3 space, Problem 3

Geometry of linear systems; incidence between lines and planes

Incidence 1: points and planes
Incidence 2: planes and lines
Incidence 3: points and lines
Parallel and orthogonal objects
Parallel planes, Problem 4
Parallel planes: Problem 5
Orthogonal planes: Problem 6
Planes and lines, Problem 7
Planes and lines, Problem 8
Planes, Problem 9
Planes, lines, and systems of equations, Problem 10

Distance between points, lines, and planes

Distances between sets, generally
Distance between points and planes
Distance between points and planes, Problem 1
Distance between points and planes, Problem 2
Distance between points and lines
Distance between points and lines, Problem 3
Distance between points and lines, Problem 4
Distance between (skew) lines
Distance between (skew) lines, Problem 5
Distance between (skew) lines, Problem 6

Some words about the next course

Linear Algebra and Geometry 1, Wrap-up
Linear Algebra and Geometry 2, some words about it
Final words

Screenshots

Linear Algebra and Geometry 1 - Screenshot_01Linear Algebra and Geometry 1 - Screenshot_02Linear Algebra and Geometry 1 - Screenshot_03Linear Algebra and Geometry 1 - Screenshot_04

Reviews

Razor
September 24, 2023
Amazing course, this helpedI me a lot and it broadened my understanding. If I were to chose between an university course and this one , I'd definitely buy this course instead
Ravi
August 13, 2023
The concepts are eloquently explained and therefore, gives clarity on subject matter. I found really a good mathematics teacher on Udemy.
Ezgi
August 1, 2023
I have followed Professor Wehlou's LAG1,LAG2 and LAG3 course for my Linear Algebra and Geometry class at college this semester and I scored full-mark + honors at my final exam while I was aiming to only pass in the beginning. Hania is an amazing teacher by teaching everything one by one and answering questins within such a short time. Thanks to her, I raised my GPA highly :)
Francisco
July 16, 2023
A very complete course in linear algebra, with a lot pedagogical resources like examples in Geogebra. Furthermore almost every explanation was completed with geometry insights and also with computations
Toby
May 30, 2023
I love it ... absolutely the best conceptual learning experience I have had. Her presentation is the best.
John
May 9, 2023
This is a 6 stars course. I'm a student a top Latin America (and world) university and I could only dream my Linear Algebra teacher had a third of the commitment and a tenth of the effort put in this course for our class. It's huge, but it means it's complete, and so well structured you can just pick what you want if you wish to with no harm whatsoever - the risk being that you miss something that you didn't know you hadn't learned. Everything is in place, cleanly and thoroughly explained, and in a way that the topics are connected and make sense when you think about the big picture. Bought it on sale because Udemy is on a budget for me, but intend to contribute full price eventually. I'll probably just copy and paste this review for the rest of Hania's courses in time.
Lisa
April 11, 2023
These courses are everthing I was looking for and even more. Up to this point I was only interested in math, but now I'm starting to love (and really unterstand!) it. Thanks for this wonderful opportunity.
Andrea
April 4, 2023
I studied linear algebra a long time ago at university and took this course simply to refresh my knowlegde. But this course didn't only refresh my knowledge and helped me to improve my rusty skills to manually compute things, I understood concepts and saw connections that I never really understood or saw before. It was all in all a wonderful experience. The lectures are very clear, extremly well structured and organized, easy to follow and fun to watch. I absolutely recommend this course. Thanks a lot for this wonderful couse. I will start the second course today and am really looking forward to it.
Hrefna
February 22, 2023
Excelente elección, la ruta de aprendizaje es buena y la explicación de los temas es adecuada. Tomaré más cursos. :3
Teenapong
January 15, 2023
I know many things in this course which I never knew before. The instructor dives into the detail with clearly explanation.
Ana
January 8, 2023
Hania is the best teacher, the courses are really amazing and all of them help to understand the reason for everything in math, she is a really good and admirable teacher, the material is incredible and the way of teaching is unique. All math teachers must watch Hania's courses before becoming math teachers. I feel very happy for finding Hania's courses, waiting for all the future ones!
Maurizio
December 26, 2022
L’Insegnante spiega con calma e precisione, approfondendo tutti gli argomenti e fornendo sempre degli esempi appropriati.
Tyrone
December 22, 2022
Thanks for taking the time to construct your courses - It's been 20 years since doing a comprehensive course in LinAlg (Applied Math and CompSci grad). And now refreshing my knowledge for those at the office who think Machine Learning etc is just a matter of running a program. My only point on Section 2 is to break the Linear Transformations section up a bit more why: This is typically one of the hardest concepts for people new to LinAlg to get their head around along with Linear combinations - both fundamental to the introduction of the subject. Anyhow - I have purchased your three LinAlg courses which I intend to plough through over my 4 weeks off over the Christmas period. It's a tricky topic to deliver! Keep up the good work. Will continue to feed back. Section 3 Nice Representations of equations and related geometrys. The nexus between the 2 is often overlooked in undergrad courses and other online instructional courses on the subject.
N
October 29, 2022
Very clear, detailed and concise explanations of topics. It is clear to see that the instructor cares greatly about the learning of her students from the effort put into this course.
SAKSH
January 17, 2022
Overall a great course for someone who wants from basic to advanced level understanding of Linear Algebra and Geometry! At least you must have a high school understanding of mathematics in order to take this course otherwise at various instances you have take things for granted without getting the feel of it!

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udemy ID
5/4/2021
course created date
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