Linear Algebra

Jila, Nov 11, 2017

Linear Algebra

Table of Contents

  1. Vectors in two dimensional plane
    1. Vector addition
  2. Vector Spaces
    1. Union and intersection of vector spaces
  3. Basis
    1. Linear Algebra, Change of basis
    2. A third vector of a basis
  4. Stochastic Matrices
    1. Power method for stochastic matrices (7 iterations.)
    2. Power method for stochastic matrices ( 49 iterations.)
  5. Linear Transformations
    1. Linear Transformations Preserve Lines
    2. Rotations
  6. Norms
    1. p-norm unit balls in 2D
    2. lp-norm unit balls in 3D
  7. Gram Schmidt Process
    1. Gram Schmidt Process for two vectors.

1.

Vectors in two dimensional plane

  • 1. Vector addition

1.1.

Vector addition

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2.

Vector Spaces

  • 1. Union and intersection of vector spaces

2.1.

Union and intersection of vector spaces

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3.

Basis

  • 1. Linear Algebra, Change of basis
  • 2. A third vector of a basis

3.1.

Linear Algebra, Change of basis

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3.2.

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4.

Stochastic Matrices

  • 1. Power method for stochastic matrices (7 iterations.)
  • 2. Power method for stochastic matrices ( 49 iterations.)

4.1.

Power method for stochastic matrices (7 iterations.)

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4.2.

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5.

Linear Transformations

  • 1. Linear Transformations Preserve Lines
  • 2. Rotations

5.1.

Linear Transformations Preserve Lines

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5.2.

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6.

Norms

  • 1. p-norm unit balls in 2D
  • 2. lp-norm unit balls in 3D

6.1.

p-norm unit balls in 2D

Here is the progression of unit ball in p-norms as p increases
There are some computational issues and the shapes are not as rounded as they should be. You can see that passed the 11-norm, it feels like the shapes are the same. That is partly computational issues with the system. Any way as p grows, the p-norm unit ball gets closer and closer to the filled square [math]-1\le x\le1[/math] and [math]-1\le y\le1[/math]. That square is the unit ball in [math]l\infty[/math]-norm.
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6.2.

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7.

Gram Schmidt Process

  • 1. Gram Schmidt Process for two vectors.

7.1.

Gram Schmidt Process for two vectors.

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