Master Matrix Algebra: The Ultimate Linear Algebra Course

Name: Master Matrix Algebra: The Ultimate Linear Algebra Course
Rating: 4.6 (9 reviews)

Learn the fundamentals you will need to understand advanced linear algebra concepts.

Created byTensor Teach

Last updated 7/2023

English

What you'll learn

Learn how to compute various properties of matrices & vectors.
Learn how to solve a system of linear equations using 3 different methods.
Learn how certain matrix operations apply to the real-world.
Develop a strong mathematical foundation for working with data.

Course content

3 sections • 39 lectures • 5h 17m total length

What Is A Vector?6:50
Adding Vectors8:02
Explore vector addition and its commutative and associative properties, with visual head-to-tail methods and component-wise computations using concrete examples.
Scalar Multiplication9:09
Explore scalar multiplication by scaling vectors, observing length changes and direction flips, and verify the distributive property with vectors a, b, and scalar c.
Vector Addition & Scalar Multiplication Practice Quiz
Calculating The Length of A Vector3:52
Compute the length (magnitude) of a vector from its i hat and j hat components using the Pythagorean theorem, shown with a 3–3 vector to yield sqrt(18).
Dot Product8:54
Explore the dot product of two vectors, its scalar result, and its geometric meaning through projections and orthogonality, including maximum alignment and zero when vectors are orthogonal.
Dot Product & Scalar Projections2:47
Explore how the dot product of vectors A and B equals the projection of A onto B and apply the formula (A·B)/|B| with a concrete example where |B| = sqrt(10).
Calculating Angle Between Two Vectors8:40
Vector Normalization13:15
Vector Lengths, Angles, and Dot Products

Introducing Matrices5:50
Define matrices as square or rectangular arrays of entries with m rows and n columns, using A11 notation, and view them as column or row vectors.
Matrix Addition9:18
Matrix Multiplication6:01
Properties of Matrix Multiplication4:06
Explore the three key ideas of matrix multiplication: it is not commutative, dimensionally constrained, and distributive over addition, with intuition and simple proofs for how these properties shape linear algebra.
Matrix Transpose4:03
Matrix Addition & Multiplication
Determinant of a Matrix11:23
Inverse of A 2x2 Matrix6:39
Inverse of A 3x3 Matrix14:33
Learn to find the inverse of a 3x3 matrix by building an augmented matrix with the identity and applying row operations until the left becomes the identity, yielding the inverse.
The Outer Product3:22
Determinants, Inverses, and Outer Products
Inner Product Definition8:48
Inner Product - Concrete Example19:48
Defining Inner Products
Inner Product - Length of A Vector11:49
Inner Product - Distance Between Vectors9:47
Inner Product - Angle Between Vectors12:43
Explore calculating the angle between two vectors using a generalized inner product, comparing the standard dot product and a custom inner product, yielding 18.43 degrees and 45 degrees.
Inner Product Operations
Types of Matrices8:34
Singular vs. Non-Singular Matrices3:23
Define singular and non-singular matrices by their determinants; a singular matrix has determinant zero and is not invertible, while a non-singular matrix has determinant not zero and is invertible.
Introduction to Orthogonal Matrices9:00
Orthogonal Matrices: Concrete Example - Part 16:47
Orthogonal Matrices: Concrete Example - Part 29:57
Permutation Matrices7:10
Working With Different Types of Matrices
Gram Schmidt Process: Introduction7:15
Gram Schmidt Process: Concrete Example11:37
Gram Schmidt Process

What Is A System of Linear Equations?5:00
Gaussian Elimination: Solving A System of Linear Equations11:16
Systems of Linear Equations: Gaussian Elimination Method
LU Decomposition: Building Motivation5:55
LU Decomposition: Finding U9:17
LU Decomposition: Finding L8:06
Compute the lu decomposition of A by using the inverses of elementary matrices m1 and m2 to obtain l and u, then use augmented matrix for back substitution.
LU Decomposition: Checking our Work4:19
Why Solving LUx=b is faster6:04
LU Decomposition
Cramers Rule: An Introduction5:12
Cramers Rule: A Concrete Example9:27
apply cramer's rule to a concrete three by three system, compute det(A) and the dx, dy, dz determinants, and solve for x, y, z.
Cramers Rule

Requirements

High-School Algebra

Description

Linear Algebra: Fundamentals of Matrix Algebra is designed to help you understand the fundamentals of Linear Algebra that will prepare you for more advanced courses in linear algebra.

You will learn how to perform a lot of matrix computations from scratch, which will be essential when learning more abstract concepts as well as applying these techniques to real-world datasets.

Topics covered include:

Vector Operations: Lengths, Normalization, Dot Products, Angles, Cross Products.
Matrix Operations & Types: Multiplication, Inversion, Reduced Row-Echelon Form
Systems of Equations: Gaussian Elimination, LU Decomposition, Cramers Rule

This course is intended for anyone that is currently taking a linear algebra course, pursuing a data science career, or any other career that uses linear algebra concepts.

This course will be followed up with a series on Linear Transformations & Vector Spaces, along with a course covering real-world applications. This is a pre-requisite to those courses and it is highly recommended that you complete this one first before moving on to the more advanced topics.

Ingenium Academy is an online learning platform aimed at providing best-in-class coverage of all math & science-related subjects. We pride ourselves on our breadth and depth of coverage of subjects and aim to fulfill this by continuing to produce more courses.

Who this course is for:

Aspiring Data Scientists
Actively Employed Data Scientists

Master Matrix Algebra: The Ultimate Linear Algebra Course

What you'll learn

Explore related topics

Course content

Vectors8 lectures • 1hr 1min

Matrices22 lectures • 3hr 12min

Systems of Linear Equations9 lectures • 1hr 5min

Requirements

Description

Who this course is for: