KindiakMath

Linear Algebra

Here are my write-ups on the theory of linear algebra and some non-trivial exercises therein. Enjoy!

Theory

Vector Spaces

  1. The Broad Basics of Vectors
  2. Creating New Vector Spaces
  3. The Key Ingredients of Vector Spaces

Linear Transformations

  1. The Linearity of Calculus
  2. The Peculiar Dimensions of Polynomial Space
  3. Entering the Matrix
  4. Theoretic Gaussian Elimination
  5. Defining Dimension
  6. The First Isomorphism Theorem
  7. The Square Root of -1

Determinants

  1. What is a Determinant?
  2. Unraveling the Determinant
  3. The Power of Permutations
  4. Manipulating Determinants

Eigenstuff

  1. Introductory Eigenstuff
  2. The Nuts and Bolts of Diagonalisation
  3. The Cayley-Hamilton Theorem
  4. Generalised Eigenstuff
  5. Generalised Diagonalisation

Inner Products

  1. The Magical Dot Product
  2. Generalised Perpendicularity
  3. Baby Approximation Theory
  4. We Finally Discuss Transposes
  5. Angle-Preserving Transformations
  6. The Spectral Theorems

Applied Linear Algebra

  1. Constructing the Cross Product
  2. Defending Rational Trigonometry
  3. Creating Euclidean Geometry
  4. Fourier Orthogonality
  5. Diagonalisation for Legends

Exercises

Vector Spaces

  1. Create Your Own Vector Space

Linear Transformations

  1. Euler’s Handy Formula
  2. A Shocking Trichotomy
  3. Reversing Linear Transformations
  4. Equating Ranges and Kernels

Determinants

  1. A Surprising Determinant

Inner Products

  1. A Unitarily Normal Problem

Applied Linear Algebra

  1. The Triple Combo
  2. A Painful Fourier Computation