MATH 110: Abstract Linear Algebra

Table of Contents

• Lecture 2 (August 29th, 2025)
• Lecture 3 (September 3rd, 2025)
• Lecture 4 (September 5th, 2025)
  • Conditions for Subspace
• Lecture 5 (September 8th, 2025)
  • Conditions for Direct Sum of Subspaces

Notes for Ken Ribet’s Fall 2025 course. A work in progress.

Lecture 2 (August 29th, 2025)

• What is a field? Describe the field of complex numbers.
• What axioms define a vector space?
  • Explicitly describe the distributive laws, with respect to maps.
• What is a function space?

Lecture 3 (September 3rd, 2025)

• How can polynomials be represented as a vector space of $n$-tuples?
  • What is $\bigcup_n \mathbb F^n$?
  • What is the degree of the polynomial 0?
• What is the null space of a matrix?

Lecture 4 (September 5th, 2025)

• In general, what is a subspace of a vector space?
• What is the smallest subspace containing a list of vectors?
• What is the sum of subspaces?

Conditions for Subspace

$U \subseteq V$ is a subspace of $V$ if $U$ is closed under the same addition and multiplication operations, and $U$ contains the additive identity of $V$.

Lecture 5 (September 8th, 2025)

• What is a direct sum of subspaces?
• When is a linear map injective?
• When are $X$ and $Y$ complimentary subspaces of $V$?

Conditions for Direct Sum of Subspaces

Condition. $U+V$ is a direct sum $\iff$ $0 = 0 + 0$ is the only way to represent $0$ as $u+v$, where $u \in U$ and $v \in V$.

Condition. $U+V$ is a direct sum $\iff$ $U \cap V = \lbrace 0\rbrace$.