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

Philosophy of linear algebra III - what is a vector?

• a vector is an element of a vector space. no more no less. so you can add them and scale them. but they belong somewhere. closure of their operations is also just as important as the operations themselves.
  • now given a basis, lets say, we have a specific pov (coordinates) on the space
  • if we wanna change the basis, the pov (coordinate) changes
  • but that’s just how we look at things! doesn’t really effect the “sacred space”
  • this also implies that the transformation of the pov (coordinates) must follow certain rule (must be linear transformations) that preserve the structure of the space
  • because if we can add vectors, then the addition of two vector is sacred, they must not depend on the pov
• if you want more characteristic to a vector, like length (or angles) between vectors, then you say the vector is an element of a normed vector space (or an inner product space)
  • again the change of pov must follow certain rule that preserve the structure of the space
  • because if we can measure lengths of vectors, then the lengths are sacred too, they must not depend on the pov
• this philosophy is opposite to “a vector(or tensor) is a specified tuple of numbers that follows some given transformation rule” which definite works~

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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.