Linear Algebra

Scanned course notes:

• Notes on linear algebra
• Notes on advanced algebra
• Notes on algebra

Notes on linear algebra follows {Horn & Johnson, 1990. Matrix Analysis}.

Four fundamental subspaces

For a linear transformation $A \in \mathbb{R}^{m\times n}$ with rank $r$ and singular value decomposition $A = U \Sigma V^T$, its four fundamental subspaces are summarized in the following table.

Table: Four fundamental subspaces of a linear transformation

Name	Notation	Alternative name	Alternative notation	Containing Space	Dimension	Basis
Image	$\mathrm{im}(A)$	Column space	$\mathrm{range}(A)$	$\mathbb{R}^m$	$r$	First $r$ columns of $U$
Kernel	$\mathrm{ker}(A)$	Null space	$\mathrm{null}(A)$	$\mathbb{R}^n$	$n-r$	Last $n-r$ columns of $V$
Coimage	$\mathrm{im}(A^\mathrm{T})$	Row space	$\mathrm{range}(A^\mathrm{T})$	$\mathbb{R}^n$	$r$	First $r$ columns of $V$
Cokernel	$\mathrm{ker}(A^\mathrm{T})$	Left null space	$\mathrm{null}(A^\mathrm{T})$	$\mathbb{R}^m$	$m-r$	Last $m-r$ columns of $U$

The fundamental theorem of linear algebra:

1. The kernel is the orthogonal complement of the coimage: $\mathrm{ker}(A)^\perp = \mathrm{im}(A^\mathrm{T})$
2. The cokernel is the orthogonal complement of the image: $\mathrm{ker}(A^\mathrm{T}) = \mathrm{im}(A)^\perp$

Introduction

• Introduction to linear algebra
• Euclidean Space

Eigenvalue, eigenvectors, and similarity

Jordan canonical form

• proof
• notes from Matrix Theory

Location and perturbation of eigenvalues

Miscellaneous Topics

• Polynomials (minimal)
• Quadratic form: Quadratic Equation
• Lambda matrices
• Linear Programming
• Minimax Theorem

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🏷 Category=Algebra Category=Linear Algebra