Matrices Mastery: Linear Algebra for AI & Engineering

Master Linear Algebra intuition: From vector basics to the Pseudoinverse for AI, Data Science, and Engineering

Created byMarco Gameiro

Last updated 12/2025

English

What you'll learn

Understand vectors and matrices intuitively, including their geometric meaning and role in linear transformations.
Master core matrix operations: dot and cross products, determinants, transpose, identity, and inverse matrices.
Determine when matrices are invertible and compute inverses using multiple methods, including Gauss–Jordan elimination.
Solve systems of linear equations using matrix methods and interpret solutions geometrically.
Apply matrix concepts to real-world problems in engineering, physics, computer graphics, and optimization.
Use the pseudoinverse matrix to solve overdetermined and underdetermined systems via least-squares methods.
Build a solid foundation for AI, machine learning, and data science applications involving matrices and optimization.

Course content

5 sections • 21 lectures • 2h 22m total length

Introduction & Dot Product7:41
Explore dot products, compute vector norms (square root of coordinate squares), and determine the angle between vectors using cosine, illustrated with concrete examples.
Dot Product7:27
Explore the dot product of vectors, including perpendicular vectors with zero, a vector with itself yielding norm squared, and the scaling rule (C A)·B = C(A·B) with a practical example.
Cross Product3:41
Learn how the cross product of two two-dimensional vectors equals X1 times Y2 minus X2 times Y1 and compare it with the dot product X1 times Y1 plus X2 times Y2.
Dot Product & Cross Product9:33
Explore dot product and cross product concepts through three cases, showing how vectors A, B, and C interact via symmetry, sums, and perpendicular relationships.
Magnitude9:52
Explore vector magnitudes via the norm of the cross product and dot product with the angle between A and B; learn conditions for parallelism and 3d determinant form.
Final Assignment Chapter 1

Determinant5:42
Explain the determinant form of cross product for vectors a and b, using determinant expansion and the minus sign before j, shown by a = 1i - 1j + 1k.
Determinants4:08
Explore determinants using AD − BC and form cross product A × B from i, j, k; with A_z = B_z = 0, it becomes k(A_x B_y − A_y B_x).
Rule of Sarrus8:18
Explore the rule of sarrus to compute 3x3 determinants by summing forward diagonals and subtracting backward diagonals.
Final Assignment Chapter 2
Project: The 3D Rotational Force Script

Identity Matrix4:40
Explore the identity matrix, a special matrix with zeros and ones on the main diagonal, defined in NDE dimensions; see I3 and I5 examples, and note NDE ≥ 2.
Transpose Matrix4:51
Learn how the transposed matrix swaps rows with columns, using A transpose, B, and C, and verify that a transpose of a transpose returns the original matrix.
Final Assignment Chapter 3
Project: Digital Image & Data Manipulation

Inverse Matrix2:25
Explore when an inverse matrix exists for square matrices, and learn the 2x2 case: compute A inverse as (1/det(A)) times the matrix [D -B; -C A].
Inverse Matrix & Matrix Multiplication9:12
Explore inverse matrices and matrix multiplication, deriving the inverse of A as 1/(ad-bc) [d, -b; -c, a], and verify A times its inverse equals the identity matrix, with N-dimensional generalization.
Assignment Number 1 - Chapter 4
Inverse Matrix & Cofactor Matrix5:30
Learn to compute a matrix inverse using the adjugate method, where the adjugate is the transpose of the cofactor matrix defined by (-1)^{i+j} M_ij, revealing a chessboard sign pattern.
Generalization of the Inverse Matrix7:16
Compute minors by removing row i and column j, form cofactors with (-1)^{i+j}, transpose to get adjugate, and use A^{-1} = (1/det A) adj(A) (simplifies in 2d).
Assignment Number 2 - Chapter 4
Gauss-Jordan Method13:42
Learn Gauss-Jordan method to find a matrix inverse by augmenting with the identity and applying row operations, illustrated on a 3×3 example and inverse verification.
Properties of the Inverse Matrix 112:41
Explore the properties of the 2x2 inverse matrix, derive the inverse formulas, and demonstrate that the inverse of a product equals the product of inverses in reverse order.
Assignment Number 3 - Chapter 4
Properties of the Inverse Matrix 25:11
Explore how to compute the transpose and inverse of a matrix, and verify that the inverse of the transpose equals the transpose of the inverse through a concrete example.
Properties of the Inverse Matrix 35:18
Compute the inverse of a 2x2 matrix A and show that scaling A by a nonzero K yields (K A)^{-1} = (1/K) A^{-1}.
Assignment Number 4 - Chapter 4
Properties of the Inverse Matrix 43:20
The lecture uses Gauss-Jordan to invert a 3x3 identity matrix and shows that its inverse remains the identity, illustrating properties of the matrix A and its inverse.
Properties of the Inverse Matrix 52:10
Show that for a symmetric matrix, its inverse is also symmetric by applying (A^{-1})^T = (A^T)^{-1} and the fact that A^T = A.
Assignment Number 5 - Chapter 5
Final Assignment Chapter 4
Project: The Smart Grid Diagnostic

Requirements

Basic pre-university mathematics (algebra and functions) is helpful but not strictly required.

Description

Unlock the power of Linear Algebra for adult learners ready to deepen their intuition or master the foundations of modern technology.

Welcome to my course! Unlock the potential of matrices with a step-by-step journey through vectors, determinants, inverses, and pseudoinverses, explained clearly and intuitively. This comprehensive course is designed for adult learners who want to strengthen their mathematical foundations, prepare for advanced STEM education, or gain the technical confidence needed for data-driven careers.

Whether you are revisiting these concepts for career growth in Engineering, preparing for university-level Data Science, or simply want to understand the mechanics of the digital world, this course will guide you through the essentials of Linear Algebra in a structured, supportive way. You will not only master the calculations but also see how matrix transformations connect to real-world problem solving and the core of modern Artificial Intelligence systems.

What You’ll Learn:

Master Vector Operations: Understand addition, scalar multiplication, and the geometric intuition of vectors.
Analyze Determinants: Learn to calculate and interpret determinants to understand how matrices scale space.
Navigate Matrix Properties: Master the roles of Identity and Transpose matrices in mathematical transformations.
Solve with Inverses: Develop a deep understanding of Matrix Inversion and the specific conditions for invertibility.
Explore the Pseudoinverse: Learn to handle non-square matrices and "unsolvable" systems - a critical skill for AI and Data Science.
See the AI Connection: Understand how these algebraic principles underpin machine learning and data processing.

Who This Course Is For ?

Adult learners preparing for higher education or refreshing core technical math skills.
Aspiring Data Scientists and AI enthusiasts who want to understand the math behind the algorithms.
Engineering students looking for a clear, intuition-first review of matrix operations.
Professionals in STEM who want to bridge the gap between abstract math and professional application.
Anyone who wants a fresh start with Linear Algebra through a clear and structured teaching style.

What’s Included:

Clear, easy-to-follow video lessons focused on intuition and clarity.
Quizzes and exercises to reinforce your mastery of matrix operations.
Downloadable PDFs for quick reference, notes, and extra practice.
Real-life insights, including how these concepts apply to modern AI and Engineering.
Lifetime access and certificate of completion to showcase your new skills.

Start learning today and see how mastering Matrices and Linear Algebra can transform your confidence, open doors in tech, and help you understand the math that powers the future of AI.

Who this course is for:

This course is designed for anyone who wants to build a strong and intuitive understanding of matrices and linear algebra. It is especially valuable for: Learners interested in understanding matrix concepts beyond formulas and memorization Engineers and technical professionals who want a clearer grasp of matrix operations People preparing for work or further study in AI, data science, robotics, or engineering Self-learners looking to strengthen their mathematical foundations Anyone who wants to understand how matrices are used in real-world applications This course is not focused on abstract theory, but on intuition, geometric meaning, and practical understanding.

Matrices Mastery: Linear Algebra for AI & Engineering

What you'll learn

Explore related topics

Course content

Fundamental Vector Operations5 lectures • 38min

Foundations of Determinants3 lectures • 18min

Essential Matrix Concepts: Identity & Transpose2 lectures • 10min

Matrix Inversion: Concepts, Methods, and Properties10 lectures • 1hr 7min

Inverse vs. Pseudoinverse Matrix1 lecture • 10min

Requirements

Description

Who this course is for: