Linear Algebra

Notes from: Linear Algebra Done Right

Vector Spaces

${\mathbb{R}}^n$ and ${\mathbb{C}}^n$

• Complex Numbers
• Lists
• n-dimensional Spaces
• Fields

Definition of Vector Spaces

• Vector Space
• Vector Spaces as Sets of Functions

Subspaces

• Subspaces
• Sums of Subspaces
  • Sum of Subspaces is the Smallest Containing Subspace
• Direct Sums of Subspaces
  • Condition for Direct Sums
  • Condition for Direct Sum of Two Subspaces

Finite-Dimensional Vector Spaces

Span and Linear Independence

• Linear Combinations
• Span
• Finite-Dimensional Vector Space
• Polynomial
• Linear Independence
• Linear Dependence Lemma
  • Length of Linearly Independent List ≤ Length of Spanning List

Bases

• Basis
  • Every Spanning List Contains a Basis
  • Basis of Finite-Dimensional Vector Space
  • Every Linearly Independent List Extends to a Basis
  • Every Subspace of V is a part of a direct sum equal to V

Dimension

• Dimension
• Dimension of a Subspace
  • Linearly Independent List of the Right Length is a Basis
  • Subspace of Full Dimension Equals the Whole Space
  • Spanning List of the Right Length is a Basis
• Dimension of a Sum
• Number of Elements in Sets vs Vector Spaces

Linear Maps

Vector Space of Linear Maps

• Linear Map
• Linear Map Lemma
• Addition and Scalar Multiplication on Sets of Linear Maps
• Products of Linear Maps
• Linear Maps Take 0 to 0

Null Spaces and Ranges

• Null Space
  • Null Space is a Subspace
• Injectivity
  • Injectivity Implies Null Space is 0 and vice versa
• Range
  • Range is a Subspace
• Surjectivity
• Fundamental Theorem of Linear Maps
• Linear Map to Lower-Dimensional Space is Not Injective
• Linear Map to Higher-Dimensional Space is Not Surjective
• Homogeneous System of Linear Equations

Matrices

• Matrix

• Matrix Representation of Linear Map

• Matrix Addition

• Scalar Multiplication of Matrix

• Matrix Multiplication

• Orthogonal Matrix

• Singular Matrix

• Permutation Matrix

• Frobenius Matrix

• Affine Transformation

• Vector Equations of Planes

Exercises

• LADR Exercises 1A
• LADR Exercises 1B
• LADR Exercises 1C
• LADR Exercises 2A
• LADR Exercises 2B
• LADR Exercises 2C
• LADR Exercises 3A
• LADR Exercises 3B
• LADR Exercises 3C
• LADR Exercises 3D
• LADR Exercises 3E
• LADR Exercises 3F

Others

• Geometry of Linear Equations
• Vector Combinations
• Matrix Basics
  • Matrix-Vector Multiplication
• Subspace Basics
• Singular Value Decomposition
• Cholesky Decomposition
• Skew-Symmetric Matrix
• Frobenius Norm