In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets – like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works.
Since we’re aiming at data-driven applications, we’ll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you’ll write code blocks and encounter Jupyter notebooks in Python, but don’t worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before.
At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these
Topic covered:
Solving data science challenges with mathematics
Motivations for linear algebra
Getting a handle on vectors
Operations with vectors
Modulus & inner product
Cosine & dot product
Projection
Changing basis
Basis, vector space, and linear independence
Applications of changing basis
Matrices, vectors, and solving simultaneous equation problems
How matrices transform space
Types of matrix transformation
Composition or combination of matrix transformations
Solving the apples and bananas problem: Gaussian elimination
Going from Gaussian elimination to finding the inverse matrix
Determinants and inverses
Einstein summation convention and the symmetry
Matrices changing basis
Doing a transformation in a changed basis
Orthogonal matrices
The Gram–Schmidt process
Gram-Schmidt process
What are eigenvalues and eigenvectors?
Special eigen-cases
Calculating eigenvectors
Changing to the eigenbasis
Eigenbasis example
Introduction to PageRank

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