Linear algebra tutorials - Rhea

The Basics

• Linear Equations and Linear Transformations
  • Introduction to Matrix
  • Matrix multiplication version 1 and version 2
  • Inverse of a Matrix
  • Introduction to Linear Transformations
• Subspaces of $ R^n $
  • Definition of Subspace
  • Definition of Kernel
  • Definition of Image
  • What is the Rank Nullity Theorem?
  • When are vectors Linearly Independent
  • When do vectors form a basis?
    • Supplementary Basis Explanations
• Orthogonality
  • When are vectors orthogonal?
• Determinants
  • What is a determinant.
  • Another intro to Determinants
  • What are some properties of the determinant?
• The inverse of a matrix

Beyond the Basics

• Dim(V) + Dim(Vorthogonal)=n
• Proof that transposing a matrix does not change its determinant
• Inner products and orthogonality
• Least Squares Solution (the formula)
• Jacobians and their Applications, by Math Squad member Joseph Ruan
• A tutorial about Bases, by Math Squad member Joseph Ruan:
  • What is a Basis? (going beyond the introductory level)
  • Basis Example Problems specifically with polynomials
  • Change of Basis
• Linear Programming

Applications

• Whitening and coloring transforms
• "divide and conquer" Discrete Fourier Transform (DFT)

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