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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.

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