Inculcation: Linear algebra

parent:: inculcation

starting out §

Philosophy of linear algebra I - finding happiness in small things (mathematical minimalism)

• We see with just the “little” definition of a vector space, we can have things like writing any vector as a unique linear combination of finitely many vectors from a smaller subset of the entire space.
• This much of structure is enough to ask a lot of questions and a solve a whole lot of problems!
• If you want more things, we can have more things! (oriented vector space, inner product spaces, normed vector spaces, etc.).

pre-first semester course essense

Start with 3b1b:

A very good place to start (basic) mathematics like linear algebra is to read chapters from Napkin:

Napkin §

Evan Chen's A Infinitely Large Napkin

• an introduction to a lots of fields of math! (NOT a textbook but a really nice introductory reference)
• starts with groups and metric spaces!

Link to original

Philosophy of linear algebra II - giving lore behind objects

• origin story of matrices: they were actually “linear maps” all along!
  • As in every matrix gives a linear map between the vector spaces $k^n\to k^m$.
  • Now try learning how to write a matrix for a given linear map between aribtrary vector spaces $V\to W$
  • Rotations are linear maps! Trace is a linear map! Transposition of matrices is a linear map! Try writing all of them as matrices (if they are not defined as matrices anyways)
  • What about derivatives tho? They are linear maps too? Can we write them as a matrix?
• origin story of tensors: they are actually multi-linear maps!
  • Maybe wait a while to start with tensors, but keep this idea in the back of your mind: tensors are multi-linear maps in a proper sense.

a first semester course §

A first semester course on linear algebra

• lectures book Linear Algebra Done Right - Sheldon Axler
  • Book website, Third ed pdf
  • YouTube lectures
• lectures Linear Algebra by Dr. K.C. Sivakumar
• Chapters 9-15 from A Infinitely Large Napkin
• mostly they cover upto canonical forms, spectral theorem

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Need help proving stuff? Try following the arrows to prove equivalent conditions for injectivity of linear maps on finite dim spaces:

Philosophy of linear algebra IV - thinking objects as part of a whole/constructing the whole first

• the set of all linear maps from $V$ to $W$ (written as $\mathsf{Hom}(V,W$ ) is made into a vector space (as a subspace of the set of all functions $V\to W$ which is also a vector space)!
• anything is possible (if it is constructable, and most things are)

after a first semester course §

• Don’t fall into traps like “geometric algebra”, it’s crap
• Tensor, symmetric and exterior algebra of a vector space

a second semester on linear algebra §

• In the first semester, you were probably in first year and only did “real” vector spaces, that is vector spaces where scalars are only coming from real numbers $\mathbb{R}$. We can generalize this to any ”field” $k$ and there are many motivations to do this.
  • $\mathbb{C}$ is algebraically closed
• The canonical forms of a linear endomorphism (fancy name for linear map $V\to V$)
• Every if you did do linear algebra on general fields in the first semester you might like to understand the intersection: category theory $\cap$ linear algebra
  • Duality is a functor $V\mapsto V^{\ast }$
  • Double duality is a natural functor $V\mapsto V^{\ast \ast }$

linear algebra after doing rings and modules §

A $k$-vector space $V$ with a endomorphism $T$ (fancy name for linear map $V\to V$) gives a $k[X]$-module structure on $V$, so we can directly use $k[X]$-modules classification theorems to construct the theorems on canonical forms.

I am still looking on how to understand two linear endomorphisms giving a $k[X,Y]$ structure on $V$. At the least, I can re-interpret the theorem that says “we always have a common eigenvector of two commuting linear maps” as the following

Let’s say we have a $k[X,Y]$-module $V$ defined by two linear endomorphisms $V\to V$. Then we always have at least one simple non-trivial $k[X,Y]$-submodule of $V$.

representation theory is just spicy linear algebra §

A group homomorphism from a (say, finite for now) group $G$ to the general linear group on a vector space $V$

$$G\to GL(V)$$

is called a representation of the group $G$. One can classify and study such homomorphisms (upto an equivalence ofcourse) and it’s called representation theory (of finite groups). This “helps” in doing linear algebra when we have a invertible linear map $V\to V$, in my opinion.