Calculus on Mengnifolds
I have no idea what happened this lecture.
Lecture highlights:
- Notations
- Review of linear algebra
- The set of bases of $V$ may be naturally identified with the set of bases of $V^\ast$
- 1-form/covector, 2-form
- Free vector spaces
- Quotient spaces
- Tensor Products
This time probably not in chronological order.
M: In my book I use the theoretical physicist convention: referring to using $u^i$ instead of $u_i$ for vector component. Our brain has very small RAM, so symbols should be as simple as possible.
M: In geometry, we don’t just consider one vector space. There are things called vector bundles, a bunch of vector spaces bundled together.
M: When I learn something, I rarely prove it at first, then when I had lots of these things, the proof will become obvious. I started to seriously study math at age of 24, when I made the transition from theoretical physics, and I think that I learn quite quickly.
M: Some peole say that physicists are algebraists, they just remember the algebraic rules. But I think that is an insult to physicists, because when they talk about physics, they have to know the meaning. They just don’t have the time to do maths.
M: There was an MIT professor called [inaudible]. He complained about tensor product. He said that now we have two algebras, algebra 1 and algebra 2. Algebra 1 is like the high end of maths and algebra 2 is like the low end. Tensor product belongs to algebra 2, so does Boolean algebra, which engineers use. Algebra 1 is like algebraic number theory and algebraic geometry.
M: If you learn linear algebra in a proper way, I’m not sure if that is the case.
M: Pointing to his universal property definition of free vector space: This should be taught in a linear algebra course, unfortunately it isn’t. This course is for prepared students. If you are not, attending these lectures will be very painful.
M: If you are serious about this course, you must spend time.
My chaotic typing
Trivialisation: an isomorphism from $V$ to $\mathbb{R}^n$. Can be identified naturally with basis.
Canonical representation of $T$ is a matrix of the form \(\begin{pmatrix} I_r & O \\ O & O \end{pmatrix}.\)
Algebra is vector space equipped with a bilinear product
Endormorphism algebra
1-form on $V$ is a linear map form $V$ to $\mathbb{R}$, also called a covector on $V$
dual space $V^\ast$ is space of covectors
$B_V \equiv B_{V^\ast}$, set isomorphism, naturally identified:
update later, or never, i dont want to decipher his whiteboard
2-form $\omega$ on $V$ is bilinear map $V^2 \to \omega \mathbb{R}$, $(\underline{u}, \underline{v} ) \to \omega (\underline{u}, \underline{v})$
$ \mathrm{Map}^{BL}(V\times V, \mathbb{R}) $ is a linear space of dim $ (\dim V)^{2} $
$ S_{2} $ acts on $ V\times V $.
A free $\mathbb{F}$-vector space over the set $X$ is a set map $\iota: X\to \mathbb{F}[x]$. $\mathbb{F}$ is a vector space, $=$ the final span/$\mathbb{F}$ of $X$.
A quotient of $V$ by $W$ is a linear map $q:V \to V/W$ with the following universal property:
The tensor product of $V_1$ with $V_2$ is a bilinear map $V_1\times V_2 \to V_1 \otimes V_2$ which satisfies the following universal property:
$\mathrm{Hom}(V_1 \times V_2, Z) \equiv \mathrm{Hom}(V_1 \otimes V_2, Z), \phi = \overline{\phi}.$
You just combine the free vector space and quotient diagrams as follows:
Notation list:
- Einstein notation: $\vec{u} = u^i \vec{e}_i $: hide the summation; to denote a single vector component, just write $u^i$
- Use $\underline{u}$ for abstract vector, and $\vec{u}$ for coordinate vector
- $B_W$ denote set of bases of $W$