Advanced Linear Algebra¶

lectur note (expired) at http://www.math.nthu.edu.tw/people/bio.php?PID=47#personal_writing

Referece¶

I. Herstein, D. Winter, A Primer on Linear Algebra.
S. Berberian, Linear Algebra.
- https://en.calameo.com/read/0004772301f1a0dda126c
J. Fraleigh, R. Beauregard, Linear Algebra.

Equivalent Classes
- well-definedness
  - independence of choice of representative
Quotients
- exercises
Direct Sum
- internal v.s. external
- ignore the difference since they are quite equal
- direct sum and Cartesian product are different when there are infinitely many of vector spaces
  - direct sum => finitely many non zero
Exact sequences
- more related to Differential Geometry and Differential Topology
- the teacher listed it for a simple preview, will not dive into it

Modules
- Modules are generalizations of vector spaces.
- Recall that a vector space is a field \(F\) and an abelian group \(V\) together with scalar multiplication \(F \times V \to V\) which satisfies (1.1). In (1.1), there is no mention of the inverse of any element of \(F\), so \(F\) can simply be a ring with 1. In that case we call \(V\) a module over the ring \(F\).
- example:
  - \(\mathbb{Z}^n\) is a module of ring \(\mathbb{Z}\)
Algebra
- have an additional operator to vector space
- \(* : V \times V \to V\)
  - distributive: \((v+w)∗x=v∗x+w∗x\) and \(x∗(v+w)=x∗v+x∗w\)
  - take out scalar: \((cx) ∗ v = x ∗ (cv) = c(x ∗ v).\)
- note that \(*\) may not be
  - commutative
  - associative
  - do not have multicative identity
- extra note on definition of ring
- example:
  - \(M_n(F)\) with matrix multiplication
  - \(M_n(F)\) with \(A*B = AB-BA\)

Note: - quaternions H and octonions O - \(N \subset Z \subset Q \subset R \subset C \subset H \subset O\)

Last update: 2023-09-15