Linear Algebra Vignette 1a: Matrix Representation of a Linear Transformation

Questions and comments below will be promptly addressed.

Linear Algebra is one of the most important subjects in mathematics. It is a subject with boundless practical and conceptual applications.

Linear Algebra is the fabric by which the worlds of geometry and algebra are united at the most profound level and through which these two mathematical worlds make each other far more powerful than they ever were individually.

Virtually all subsequent subjects, including applied mathematics, physics, and all forms of engineering, are deeply rooted in Linear Algebra and cannot be understood without a thorough understanding of Linear Algebra. Linear Algebra provides the framework and the language for expressing the most fundamental relationships in virtually all subjects.

This collection of videos is meant as a stand along self-contained course. There are no prerequisites. Our focus is on depth, understanding and applications. Our innovative approach emphasizes the geometric and algorithmic perspective and was designed to be fun and accessible for learners of all levels.

Numerous exercises will be provided via the Lemma system (under development)

We will cover the following topics:
Vectors
Linear combinations
Decomposition
Linear independence
Null space
Span
Linear systems
Gaussian elimination
Matrix multiplication and matrix algebra
The inverse of a matrix
Elementary matrices
LU decomposition
LDU decomposition
Linear transformations
Determinants
Cofactors
Eigenvalues
Eigenvectors
Eigenvalue decomposition (also known as the spectral decomposition)
Inner product (also known as the scalar product and dot product)
Self-adjoint matrices
Symmetric matrices
Positive definite matrices
Cholesky decomposition
Gram-Schmidt orthogonalization
QR decomposition
Elements of numerical linear algebra

I’m Pavel Grinfeld. I’m an applied mathematician. I study problems in differential geometry, particularly with moving surfaces.

[gentlemandude]

Your contribution to students of linear algebra is immeasurable. Thank you for all of your hard work over the years producing these exceptional videos.

[PatrickEngSU]

These videos are great. I wish they were around 14 years ago when I was first learning linear algebra

[AbrahamHoffman]

I am starting my journey to learn linear algebra with this video! Thanks so much for providing it.

[NTC]

OMG another AWESOME free lecture series. Thank you very much for posting this. You're the second best lecturer I've seen. I would say the best one is Gilbert Strang.

[davidcabreramartinez]

Hello, Minute 22 I didn’t get why “Re2” is [-2 1], thank you.

[DanBrickley]

Prior to choosing our basis, are we nevertheless assuming that our vectors live in R2? Hard to be sure on a blackboard, but more deeply it wasn't immediately clear if the dimensionality is basis independent.

[untwerf]

I didn't catch the name of the author of the second linear algebra book u mention... u said Gilbert strang but the second author's name sounded like "gelfon" or "galfond"... would u mind confirming the second author's name?!