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Master linear algebra: theory and implementation in code

Name: Master linear algebra: theory and implementation in code
Rating: 4.6 (6966 reviews)

Learn concepts in linear algebra and matrix analysis, and implement them in MATLAB and Python.

Created byMike X Cohen

Last updated 6/2026

English

What you'll learn

Understand theoretical concepts in linear algebra, including proofs
Implement linear algebra concepts in scientific programming languages (MATLAB, Python)
Apply linear algebra concepts to real datasets
Ace your linear algebra exam!
Apply linear algebra on computers with confidence
Gain additional insights into solving problems in linear algebra, including homeworks and applications
Be confident in learning advanced linear algebra topics
Understand some of the important maths underlying machine learning
The math underlying most of AI (artificial intelligence)

Course content

16 sections • 171 lectures • 33h 46m total length

What is linear algebra?8:03
Get a broad overview of linear algebra
Linear algebra applications5:57
Learn about some applications of linear algebra
An enticing start to a linear algebra course!12:01
How best to learn from this course3:59
Maximizing your Udemy experience7:57

Algebraic and geometric interpretations of vectors12:45
Learn two ways of interpreting vectors (this is the "algebraic-geometric dualist perspective" in linear algebra).
Vector addition and subtraction8:26
How to do basic arithmetic with vectors.
Vector-scalar multiplication9:07
Multiply a vector by a number, and learn why "scalars" are called scalars.
Vector-vector multiplication: the dot product10:11
Arguably the most important and fundamental computation in all of linear algebra!
Dot product properties: associative, distributive, commutative18:55
Learn several important properties of the vector dot product.
Code challenge: dot products with matrix columns8:17
Use a for-loop to compute dot products between corresponding columns.
Code challenge: is the dot product commutative?9:28
Is the dot product commutative? Use a computer to find out!
Vector length6:42
Learn to compute the length of a vector.
Vector length in MATLAB
Vector length in Python
Dot product geometry: sign and orthogonality23:38
How to interpret the sign of a dot product from a geometric perspective.
Vector orthogonality
Code challenge: Cauchy-Schwarz inequality17:34
Relative vector angles
Code challenge: dot product sign and scalar multiplication12:05
Implement what you learned in code!
Vector Hadamard multiplication3:43
Learn the "sensible way" to multiply two vectors.
Outer product10:17
Create a matrix from two vectors using the outer product.
Vector cross product9:05
The special multiplication for 3-D vectors.
Vectors with complex numbers8:17
Learn the basics of complex numbers and complex vectors.
Hermitian transpose (a.k.a. conjugate transpose)16:21
If you ever work with complex numbers in linear algebra, you need to know about the Hermitian!
Interpreting and creating unit vectors7:58
"Normalize" a vector by giving it length=1.
Code challenge: dot products with unit vectors13:33
Dimensions and fields in linear algebra7:54
Important linear algebra terminology.
Subspaces15:50
A subspace is an important concept in linear algebra that is fundamental for many other topics.
Subspaces vs. subsets5:47
Two very different but easily confused topics.
Span13:29
Learn the algebraic and geometric interpretations of a span of a set of vectors.
In the span?
Linear independence15:34
The linear algebra declaration of linear independence!
Basis11:51
Combine independence and basis into one concept.

Matrix terminology and dimensionality8:14
Learn the basic terminology of matrices.
Matrix sizes and dimensionality
A zoo of matrices17:19
Many matrices are given special names, here are some of them.
Can the matrices be concatenated?
Matrix addition and subtraction8:28
Basic arithmetic with matrices.
Matrix-scalar multiplication2:33
Multiply a matrix by a number.
Code challenge: is matrix-scalar multiplication a linear operation?7:28
Use computers to test whether u(A+M) = uA+uM
Transpose10:24
Flipping off a matrix or vector is actually a good thing in linear algebra.
Complex matrices1:51
What you learned with complex vectors applies to matrices.
Addition, equality, and transpose
Diagonal and trace9:07
How to work with the diagonal elements of a matrix.
Code challenge: linearity of trace9:37
Apply your knowledge to learn a new concept in linear algebra.
Broadcasting matrix arithmetic14:13

Introduction to standard matrix multiplication10:27
Matrix multiplication gets its own introduction.
Four ways to think about matrix multiplication11:55
Strange but true: There are four different ways to think about matrix multiplication.
Code challenge: matrix multiplication by layering9:45
Implement matrix multiplication in code.
Matrix multiplication with a diagonal matrix3:42
Diagonal matrices are convenient for many reasons, including simplicity of multiplication.
Order-of-operations on matrices8:15
Learn the "LIVE EVIL" rule!
Matrix-vector multiplication16:43
Key properties of matrix-vector multiplication.
Find the missing value!
2D transformation matrices15:32
A geometric interpretation of matrix-vector multiplication.
Code challenge: Pure and impure rotation matrices12:38
Code challenge: Geometric transformations via matrix multiplications15:58
Also, gain new insight into the meaning of singular values!
Additive and multiplicative matrix identities6:19
Two key matrix identities lead to the zero matrix and the identity matrix.
Additive and multiplicative symmetric matrices15:16
Learn how to create symmetric matrices.
Hadamard (element-wise) multiplication5:00
Yet another way to multiply matrices.
Matrix operation equality
Code challenge: symmetry of combined symmetric matrices12:03
Use code to learn an important concept in linear algebra.
Multiplication of two symmetric matrices13:21
Is the product of two symmetric matrices symmetric? Find out!
Code challenge: standard and Hadamard multiplication for diagonal matrices6:27
Use code to learn a special property of multiplication with diagonal matrices.
Code challenge: Fourier transform via matrix multiplication!11:20
Create a Fourier matrix and implement the Fourier transform.
Frobenius dot product11:16
The Frobenius dot product is used often in statistics and machine learning.
Matrix norms18:11
Learn several commonly used matrix norms.
Code challenge: conditions for self-adjoint11:52
Code challenge: The matrix asymmetry index29:02
What about matrix division?4:24
Conceptual and implementational aspects of matrix division.

Rank: concepts, terms, and applications10:50
Learn the key properies and uses of matrix rank.
Maximum possible rank.
Computing rank: theory and practice23:00
Learn the distinction between rank in a first-year course vs. rank in real-world applications.
Rank of added and multiplied matrices11:46
Upper bounds of the ranks of added and multiplied matrices.
What's the maximum possible rank?
Code challenge: reduced-rank matrix via multiplication10:38
Create an MxN matrix with rank r.
Code challenge: scalar multiplication and rank12:10
Does scalar multiplication change the rank of a matrix? Test your hypothesis in code!
Rank of A^TA and AA^T10:41
Rank of our favorite type of matrix: A^TA.
Code challenge: rank of multiplied and summed matrices7:06
Use code to confirm your theoretical knowledge.
Making a matrix full-rank by "shifting"14:12
Transform a rank-deficient to a full-rank matrix using this one simple trick!
Code challenge: is this vector in the span of this set?11:46
Use code to test span.
Course tangent: self-accountability in online learning3:03

Column space of a matrix13:29
Apply a concept you've already learned (span) to a new domain.
Column space, visualized in code6:35
The column space of a matrix, visualized in MATLAB.
Row space of a matrix4:25
Really, it's the same as the column space, but transposed.
Null space and left null space of a matrix14:39
Learn how to interpret and find the "null space" of a matrix.
Column/left-null and row/null spaces are orthogonal10:47
Learn some interesting features of matrix spaces.
Dimensions of column/row/null spaces8:10
See how the puzzle pieces of matrix spaces all fit together.
Example of the four subspaces11:09
See an example of extracting bases for the four subspaces of a matrix.
More on Ax=b and Ax=07:52
These equations look simple, but they are really important for applied linear algebra.

Systems of equations: algebra and geometry19:39
Understand the algebra and geometry and systems of equations.
Converting systems of equations to matrix equations4:23
Converting equations into matrices is the first step of advanced statistics and model-fitting.
Gaussian elimination14:42
Use Gaussian elimination to solve a system of equations.
Echelon form and pivots7:21
Learn how to recognize and compute the echelon form of a matrix, and how to obtain the pivots.
Reduced row echelon form18:29
Compute the reduced row echelon form and see how the RREF is used in solving systems of equations.
Code challenge: RREF of matrices with different sizes and ranks12:16
See the RREF form of difference kinds of matrices.
Matrix spaces after row reduction9:23
How does RREF affect matrices spaces? Watch to find out!

Determinant: concept and applications5:59
Learn the key properies and uses of matrix rank.
Determinant of a 2x2 matrix7:03
Learn the shortcut to compute the determinant of a 2x2 matrix.
Code challenge: determinant of small and large singular matrices11:07
See the difference between determinant in theory and determinant in computer applications.
Determinant of a 3x3 matrix13:13
Learn the shortcut to compute the determinant of a 2x2 matrix.
Code challenge: large matrices with row exchanges6:32
See what happens when you swap rows of larger matrices.
Find matrix values for a given determinant4:51
A different perspective on the determinant will lead the way towards discovering eigenvalues!
Code challenge: determinant of shifted matrices18:27
Discover the effects of "shifting" a matrix on its determinant.
Code challenge: determinant of matrix product10:37
Illustrate a special property of the determinant, while also showing how unstable the determinant can be for larger matrices!

Matrix inverse: Concept and applications12:40
Learn the key concepts and uses of the matrix inverse.
Computing the inverse in code6:31
See how the inverse is computed and visually represented.
Inverse of a 2x2 matrix7:55
There is a handy short-cut for inverting a 2x2 matrix!
The MCA algorithm to compute the inverse13:58
This is the full algorithm to compute the inverse of any invertible matrix.
Code challenge: Implement the MCA algorithm!!18:39
You know the theory, now make the magic happen in MATLAB (Or Python)!
Computing the inverse via row reduction16:40
Another way to compute the inverse is RREF, which has a more intuitive explanation compared to the MCA procedure.
Code challenge: inverse of a diagonal matrix10:50
Use code to discover an interesting property of inverses of diagonal matrices.
Left inverse and right inverse10:14
Think a rectangular matrix can't be inverted? Think again!
One-sided inverses in code12:40
See how the one-sided inverse is implemented in MATLAB.
Proof: the inverse is unique3:16
One inverse to rule them all!
Pseudo-inverse, part 111:34
A gentle introduction to the Moore-Penrose pseudoinverse.
Code challenge: pseudoinverse of invertible matrices6:02
Find out what happens when you compute the pseudoinverse of an invertible matrix.
Why should you avoid the inverse?0:14

Requirements

Basic understanding of high-school algebra (e.g., solve for x in 2x=5)
Interest in learning about matrices and vectors!
(optional) Computer with MATLAB, Octave, or Python (or Jupyter)

Description

You need to learn linear algebra!

Linear algebra is perhaps the most important branch of mathematics for computational sciences, including machine learning, AI, data science, statistics, simulations, computer graphics, multivariate analyses, matrix decompositions, signal processing, and so on.

You need to know applied linear algebra, not just abstract linear algebra!

The way linear algebra is presented in 30-year-old textbooks is different from how professionals use linear algebra in computers to solve real-world applications in machine learning, data science, statistics, and signal processing. For example, the "determinant" of a matrix is important for linear algebra theory, but should you actually use the determinant in practical applications? The answer may surprise you, and it's in this course!

If you are interested in learning the mathematical concepts linear algebra and matrix analysis, but also want to apply those concepts to data analyses on computers (e.g., statistics or signal processing), then this course is for you! You'll see all the maths concepts implemented in MATLAB and in Python.

Unique aspects of this course

Clear and comprehensible explanations of concepts and theories in linear algebra.
Several distinct explanations of the same ideas, which is a proven technique for learning.
Visualization using graphs, numbers, and spaces that strengthens the geometric intuition of linear algebra.
Implementations in MATLAB and Python. Com'on, in the real world, you never solve math problems by hand! You need to know how to implement math in software!
Beginning to intermediate topics, including vectors, matrix multiplications, least-squares projections, eigendecomposition, and singular-value decomposition.
Strong focus on modern applications-oriented aspects of linear algebra and matrix analysis.
Intuitive visual explanations of diagonalization, eigenvalues and eigenvectors, and singular value decomposition.
Improve your coding skills! You do need to have a little bit of coding experience for this course (I do not teach elementary Python or MATLAB), but you will definitely improve your scientific and data analysis programming skills in this course. Everything is explained in MATLAB and in Python (mostly using numpy and matplotlib; also sympy and scipy and some other relevant toolboxes).

Benefits of learning linear algebra

Understand statistics including least-squares, regression, and multivariate analyses.
Improve mathematical simulations in engineering, computational biology, finance, and physics.
Understand data compression and dimension-reduction (PCA, SVD, eigendecomposition).
Understand the math underlying machine learning and linear classification algorithms.
Deeper knowledge of signal processing methods, particularly filtering and multivariate subspace methods.
Explore the link between linear algebra, matrices, and geometry.
Gain more experience implementing math and understanding machine-learning concepts in Python and MATLAB.
Linear algebra is a prerequisite of machine learning and artificial intelligence (A.I.).

Why I am qualified to teach this course:

I have been using linear algebra extensively in my research and teaching (in MATLAB and Python) for many years. I have written several textbooks about data analysis, programming, and statistics, that rely extensively on concepts in linear algebra.

So what are you waiting for??

Watch the course introductory video and free sample videos to learn more about the contents of this course and about my teaching style. If you are unsure if this course is right for you and want to learn more, feel free to contact with me questions before you sign up.

I hope to see you soon in the course!

Mike

Who this course is for:

Anyone interested in learning about matrices and vectors
Students who want supplemental instruction/practice for a linear algebra course
Engineers who want to refresh their knowledge of matrices and decompositions
Biologists who want to learn more about the math behind computational biology
Data scientists (linear algebra is everywhere in data science!)
Statisticians
Someone who wants to know the important math underlying machine learning
Someone who studied theoretical linear algebra and who wants to implement concepts in computers
Computational scientists (statistics, biological, engineering, neuroscience, psychology, physics, etc.)
Someone who wants to learn about eigendecomposition, diagonalization, and singular value decomposition!
Artificial intelligence students

Master linear algebra: theory and implementation in code

What you'll learn

Explore related topics

Course content

Introductions5 lectures • 38min

Get the course materials1 lecture • 9min

Vectors24 lectures • 4hr 37min

Introduction to matrices10 lectures • 1hr 29min

Matrix multiplications21 lectures • 4hr 9min

Matrix rank10 lectures • 1hr 55min

Matrix spaces8 lectures • 1hr 17min

Solving systems of equations7 lectures • 1hr 26min

Matrix determinant8 lectures • 1hr 18min

Matrix inverse13 lectures • 2hr 11min

Requirements

Description

Who this course is for: