Minerva CS113 Linear Algebra

Concept Notes

Unit 1: Linear Systems

• linear system
  • linear equation
  • augmented matrix
    • row operations = pre-multiply by elementary matrix
  • Gaussian elimination
• matrix algebra
  • dot product
  • matrix multiplication
  • cross product
• special matrices
  • zero matrix
  • identity matrix
  • elementary matrix
  • diagonal matrix
  • triangular matrix
• inverse matrix
  • invertible
  • singular
• transpose
  • symmetric matrix
• determinant
• trace

Unit 2: Vector Space Structure

• scalar, vector, matrix, tensor
• vector space & subspace
• linear combination & span
• linearly dependent & linearly independent
• basis & dimension
• subspace
  • fundamental subspaces
  • rank
  • null space
  • column space
• norm
  • unit vector
  • L2 norm
  • L1 norm
  • Frobenius norm
• orthogonal
  • orthonormal
  • orthonormal basis
  • orthogonal complement
  • orthogonal projections
• normal equation for linear regression

Unit 3: Transformations

• linear transformation
• change of basis
• eigenstuff
  • eigenvalue
  • eigenvector
  • eigenspace
• Markov transition matrix
• diagonalization
• the Spectral Theorem

Bookmarks

• Compiling all Sagemath code

Resources

1. Minerva CS113 - Video
2. Articles: Cliff Notes
3. Book: Linear Algebra
4. Book: Mathematics for Machine Learning
5. Book: The Matrix Cookbook - Quick references
6. 3Blue1Brown: Essence of Linear Algebra
7. fast.ai: Computational Linear Algebra

Practice

1. Albert.io
2. Book: Linear Algebra