Table of Contents

Definitions

  1. Span:

  2. Linear Independence:

  3. Subspace:
    • Definition:
    • Geometrical Interpretation:
    • Mathematical Representation:
  4. Affine Sets and Subspaces (Cosets - Abstract Algebra):
    • Definition:
    • Geometrical Interpretation:
    • Mathematical Representation:
    • Special Case of a single basis vector:

    • Find the Affine Subspace Corresponding to the following set:
      The set \(\boldsymbol{L}\) in \(\mathbb{R}^3\) defined by:
      \(x_{1}-13 x_{2}+4 x_{3}=2, \quad 3 x_{2}-x_{3}=9\)
      \(5x_{1}-8x_{2}+17 x_{3}=2, \quad 6 x_{2}-2x_{3}=13\)

    • Mathematical Representation of a line:

  5. Basis:

  6. Dimension:

Norms and Scalar Products

  1. Scalar/Inner/Dot Product:

  2. Norms:
    • Definition + Theorem (properties):

  3. \(l_p\) Norms:

  4. The \(l_1-norm\):
    • (Geometrically) Corresponds to:
  5. The \(l_2-norm\) (Euclidean Norm):
    • (Geometrically) Corresponds to:
    • Properties:
  6. The \(l_\infty-norm\):
    • (Geometrically) Corresponds to:
    • Application:

  7. The Cardinality:

  8. Cauchy-Schwartz inequality:

  9. Angles between vectors:

Orthogonality

  1. Orthogonal Vectors:

  2. Orthogonal Matrix:

Projections

  1. Line:
    • Definition:
    • Mathematical Representation:
  2. Projection on a line:
    • Set up Equation:
  3. The Projection:
    • Solve Equation:
  4. Interpreting the scalar product:

Hyperplanes

  1. Hyperplanes:
    • Two definitions:
  2. Hyper-Planes as Affine Sets:
    • How are they useful?:
  3. Geometry of Hyperplanes:

Half-Spaces

  1. Half-Space:

  2. Geometric Interptation:

Linear Functions and Transformations, and Maps

  1. Linear Functions:

  2. Affine Functions:

  3. Equivalent Definitions of Linear Functions [Theorem]:

  4. Vector Form (and the scalar product):

  5. Gradient of a Linear Function:

  6. Gradient of an Affine Function:

  7. Interpreting \(a\) and \(b\):

  8. First-order approximation of non-linear functions:

  9. First-order Expansion of a function [Theorem]:

Matrices

  1. Matrix Transpose:

  2. Matrix-vector product:

  3. Left Product:

  4. Matrix-matrix product:

  5. Block Matrix Products:

  6. Outer Products:

  7. Trace:

  8. Scalar Product:

  9. Special Matrices:

Matrix Norms

  1. Norm:

  2. \(l_{p,q}\) norms:

  3. \(l_{2,2}\) (Frobenius norm):

  4. \(l_{\infty,\infty}\) (Max Norm):

  5. The Spectral Norm:{: .bodyContents9 #bodyContents95

  6. Asynchronous:

  7. Asynchronous:

  8. Equivalence of Norms:

  9. Applications: