Home Calculus III - Lecture 2

Calculus III - Lecture 2

Calculus on Mengnifolds

I have no idea what happened this lecture.

Lecture highlights:

  1. Notations
  2. Review of linear algebra
  3. The set of bases of $V$ may be naturally identified with the set of bases of $V^\ast$
  4. 1-form/covector, 2-form
  5. Free vector spaces
  6. Quotient spaces
  7. 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:

  1. Einstein notation: $\vec{u} = u^i \vec{e}_i $: hide the summation; to denote a single vector component, just write $u^i$
  2. Use $\underline{u}$ for abstract vector, and $\vec{u}$ for coordinate vector
  3. $B_W$ denote set of bases of $W$
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