Linear algebra happy fun time with Dan and Bernardo

DanielSank

00:07

back

heather

hello

DanielSank

Ok heather, we want the ith component of $Tv$ in basis $e$.

heather

right

DanielSank

So we do $$\langle e_i | T | v \rangle$$

Does that make sense?

heather

maybe? let me look back at the earlier definition real quick

okay, yes I think that makes sense.

I think I see where I went wrong.

DanielSank

00:12

You can re-parse it as $\langle e_i | Tv\rangle$, or in other words, $\vec{e}_i \cdot \vec{(Tv)}$.

heather

Then do we do $\sum^n_{i=1}\langle e_i|T|v_i\rangle$?

DanielSank

@heather Almost. $|v\rangle = \sum v_i^e |e_i\rangle$.

heather

or just $v$ at the end there?

@DanielSank how did you get that?

DanielSank

$v$ is a vector.

We can write it as a weighted sum of the basis vectors.

(That's what it means to be a basis: any vector can be written as a sum of the basis vectors).

So, I say that $v$ is a sum over the $|e_i\rangle$'s, with weights $v_i^e$.

It's just fancy-pantsy notation for a weighted sum. I'm using this very funny looking notation because it's explicit.

heather

oh...I think I understand now.

okay.

Bernard Meurer

00:16

@DanielSank ping me when we can go back

DanielSank

ok so if you have $\langle e_i | T | v \rangle$, then insert the sum!

Write the expression with the sum inserted.

@BernardMeurer yeah ok just a few minutes.

heather

$\sum^n_{i=1}\langle e_i|T|\sum v^e_i|e_i\rangle\rangle$?

DanielSank

@heather Yep, but your notation is lightly wonky.

You don't need the final $\rangle$.

You also don't need the first $|$.

Oh wait a second, why do you have two sums!?

How did you invent a second sum?

heather

00:46

Um...

I'm good at inventing things?

I thought you said I had to plug in the sum into the other thing I wrote, which had a sum...

DanielSank

Miscommunication.

We start with $$(Tv)_i^e \equiv \langle e_i | T | v \rangle$$

That's a definition.

Now plug the sum in for $v$.

heather

okay, the it'd be $\langle e_i|T|\sum v^e_ie_i\rangle$

right?

DanielSank

Yes!

heather

=D

DanielSank

Ok, now do you know about how to deal with sums in dot products?

For example: $$\vec{v} \cdot (\vec{a} + \vec{b})$$

heather

00:59

would you add a and b and then do v $\cdot$ the added a and b

DanielSank

Could do that, yep.

heather

but I guess you can't do that here

is it maybe distributive? like v $\cdot$ a + v$\cdot$ b

DanielSank

However, you can also do $$\vec{v} \cdot (\vec{a} + \vec{b}) = \vec{v} \cdot \vec{a} + \vec{v} \cdot \vec{b}$$

Yes.

heather

okay

what is the interval for that summation?

DanielSank

01:14

Interval?

Oh, $\sum_{i=1}^n$.

heather

sorry, I don't know the correct term for it.

okay

DanielSank

i.e. there are $n$ basis vectors.

heather

oh, that makes sense

DanielSank

(btw, the point of all this is that you're about to derive the rule of matrix multiplication)

heather

$\langle e_i|T\sum^n_{i=1} v^e_ie_i\rangle$

DanielSank

01:16

Yep, so far so good...

What can you remove from the dot product?

i.e. what can you factor out?

heather

$e_i$

?

DanielSank

Well, let's write it out...

heather

okay, one moment

$=\langle e_i|T(v_1^e+...+v_n^e)(e_1+...+e_n)\rangle$

maybe those parentheses aren't right...not sure there

DanielSank

That looks suspicious. If I multiply out the terms in parentheses I get all kinds of stuff like $v_1^e e_4$.

You have this: $$\langle e_i | T (v_1^e |e_1\rangle + v_2^e |e_2\rangle + \cdots)$$

heather

oh, one quick side note: @Skyler wants to be allowed to talk here =) a bribe is offered of some notes specifically geared to what we're talking about @DanielSank

DanielSank

01:20

But written as a sum it's $$\langle e_i | T \sum_{j=1}^n v_j^e e_j \rangle$$

Bernard Meurer

I'm literally jealous

Skyler

yo

heather

yeah, that does seem to be wrong

@BernardMeurer, sorry!

Skyler

lets see if I can get my Phys 102 notes

heather

hmm, could you write it as $=\langle e_i|T(v_1^ee_1+...+v_n^ee_n)\rangle$?

Skyler

01:21

day 1 had a really good review iirc that can set up the fundamentals

DanielSank

@heather Yes.

heather

@DanielSank, yay =)

DanielSank

So now you can express this as a sum of dot products...

Skyler

btw, have you started on the path of calculus as well as linear algebra @heather ?

just might change what I send Dan's way

heather

@Skyler, yes, I have; I've been practicing single-variable, and hope to start learning multi-variable calc over Christmas break =)

Calculus is cool

DanielSank

01:24

@Skyler FYI I'm working on a book on this subject.

Skyler

@heather you're cool

Bernard Meurer

Okay folks, I'm heading to bed

heather

so $e_i\cdot v_1^ee_1+...+e_i\cdot v_n^ee_n$?

Skyler

makes me wish I had found Physics SE back in high school

@DanielSank Ever heard of Physics 102 at our school?

Bernard Meurer

@DanielSank My next algebra class is on thursday, so if you had time I want to try again. I'm trying to make a nice solution using matrices to show my prof because she asked

Anyway, another time, goodnight

DanielSank

01:25

@heather Yes, now write that with $\Sigma$ notation.

heather

@Skyler physics.SE is awesome =)

DanielSank

@BernardMeurer Ok cool!

heather

@BernardMeurer, I'm sorry, you should've gone first in the question asking =/

Have a good night

Skyler

We had this teacher who couldn't really teach the material but literally wrote out a books worth on the board, if you copied it you basicall had 85% of a book done

heather

$\sum^n_{i=1}e_i\cdot v_i^ee_i$?

DanielSank

01:27

What happened to $T$? :-P

heather

ah, yes, the vanishing $T$ =P

let me fix that

DanielSank

@heather Oh yeah, $T$ is missing here.

Skyler

Hopefully later tonight I'll have two sets of notes to send your guys's way

heather

yep, it is.

let me fix both of those =P

$T(e_i\cdot v_1^ee_1+...+e_i\cdot v_n^ee_n)$ for the first part?

$T\sum^n_{i=1}e_i\cdot v_i^ee_i$ for the second part?

DanielSank

@heather That doesn't quite work. Dot products are numbers and a matrix acting on a number isn't a thing.

You pulled $T$ out farther than allowed.

Right idea though!

Very close.

heather

01:28

@DanielSank, that makes sense, okay

hmm.

maybe, $T$ times each individual part? so:

DanielSank

@heather Yes.

Well maybe.

Depends what you mean.

heather

$T(e_i\cdot v_1^ee_1)+...+T(e_i\cdot v_n^ee_n)$ for the first part?

DanielSank

You still have $T$ acting on numbers.

heather

no yeah, that's wrong

whoops

$e_i\cdot T(v_1^ee_1)+...+e_i\cdot T(v_n^ee_n)$

for the first part?

and $\sum^n_{i=1}e_i\cdot T(v_i^ee_i)$ for the second?

DanielSank

@heather YES!

You can make one more simplification...

$v_i^e$ are numbers. They can come outside of the dot product.

heather

01:33

really? wait, it's $e_i\cdot e_i$?

DanielSank

@heather Oh no! We made a dumb mistake: used the same variable for the left vector and for our sum.

Let's use a different index for the sum.

Silly mistake on our part.

Very common.

heather

okay, let's make it $e_x$ (does it matter what you pick?)

DanielSank

Use $j$.

heather

okay

$e_j$

or $\sum^n_{i=1}e_j\cdot T(v_i^ee_i)$

and then you can pull out the $v^e_i$

DanielSank

Yah.

heather

01:36

so you get $\sum^n_{i=1}v_i^e(e_j\cdot T(e_i))$?

DanielSank

Yes.

Now, if you don't mind, I'm going to write exactly the same thing but in bra-ket notation.

$$\sum_{i=1}^n \langle e_j | T | e_i \rangle v_i^e$$

heather

okay

so...hmm, okay, I guess that makes sense, yeah

yeah, that makes sense

DanielSank

Now, suppose we arrange those numbers $\langle e_j | T | e_i \rangle$ into a grid.

Call each one $T^e_{ji}$...

Then we have $$(Tv)^e_j = \sum T^e_{ji} v_i^e$$

Look familiar?

heather

wait, wait, sorry, rewind...

@DanielSank this part is confusing to me

how do you get that?

DanielSank

@heather I just define new notation: $$T_{ji}^e \equiv \langle e_j | T | e_i \rangle$$

It's just shortcut notation, absolutely nothing more.

heather

01:40

sorry brb

DanielSank

kk

heather

01:58

okay, I'm back

sorry about that

okay, shortcut notation that you defined

okay, that makes sense.

DanielSank

It's just notation.

heather

right

DanielSank

Let's say our equation expression in English:

The jth component of $Tv$ in basis $e$ is found by summing over the jth row of $T^e_{ji}$ multiplied by the components of $v$ in basis $e$.

It's literally matrix multiplication!

heather

wow

DanielSank

Do you see it?

heather

02:02

yeah, it is!

DanielSank

Ok, good. So what have we learned?

Matrix multiplication exists because it answers the following question: What are the components of a transformed vector in terms of the components of the original vector and the elements of the matrix representing that transformation?

heather

1. bra-ket notation is not super-scary

DanielSank

Yes.

heather

2. neither are summations

DanielSank

Yes.

$T^e_{ji}$ are called matrix elements of the transformation $T$ in the basis $e$.

heather

02:04

@DanielSank wow...yeah, i'd almost forgotten what our starting problem was!

DanielSank

:-)

I had you do this so you can see a deeper meaning to matrix multiplication. It's not just some random operation. It's how you compute components of vectors that have been transformed (once you pick a basis!).

heather

and that really makes a ton of sense! it's basically answering what rule you can follow to switch between the transformed and non-transformed vectors! wow...

linear algebra is so deep

DanielSank

It's surprisingly rich for how simple the basic elements are.

heather

wow...

I'm still kind of staring at it =P

@DanielSank, thanks so much for walking me through this (and putting up with some of my ridiculous answers)

DanielSank

@heather No problem.

This is a very important topic.

I want people to understand it.

heather

02:08

@DanielSank, I can't wait for your book

DanielSank

@heather Working on it (slowly).

heather

if you write about topics like you explain them here, then you could become a millionaire textbook writer =)

DanielSank

The first chapter builds if you clone the git repo, but it's in the middle of a revision...

@heather We'll see.

heather

I didn't realize there was a git repo for it

DanielSank

@heather Yes ma'am.

heather

02:10

hmm, I don't see it on your github profile...is it in the theory repo?

DanielSank

It has a misleading name :-P

github.com/DanielSank/FourierTransform

It has a submodule, so do

heather

oh, okay =D

DanielSank

$ git clone --recursive name_of_repo

I should put that in the readme...

heather

i put the url in for the name of the repo, right?

DanielSank

Yes.

I'm working on cleaning up the whole thing. I'm happy with most of the structure but the very first part is a bit badly written and the figures need work.

heather

02:13

okay, cloned it

DanielSank

:-D

If you'll want to contribute, go ahead and make a fork.

heather

which is the main file to compile?

DanielSank

But anyway, in the finite dimensions chapter, you can build main.tex after running the python script that makes the figures.

Oh, install inkscape.

$ sudo apt-get install inkscape

It's a graphics drawing program. I use it to draw figures and convert them to pdf.

heather

installing inkscape

DanielSank

Inkscape is sweet.

heather

02:16

let's see, python script...

DanielSank

All the figures in my papers are inkscape and/or python's plotting lib.

heather

make-figures.py

okay, and then python 2 or 3 for that?

DanielSank

Probably works with either.

Skyler

@DanielSank I wonder if I'll be able to make it into my second acknowledgement chapter of a Physics textbook (tapping fingers together mischievously)

DanielSank

@Skyler Do it.

Also, try not to chat about non-linear-algebra too much in here.

I hang out in hbar too.

Skyler

02:18

Aye aye

heather

I can move over if you want @DanielSank

DanielSank

@heather You're fine, the book is related.

heather

okay =)

ah, it compiled nicely =)

DanielSank

Good.

heather

hmm, is it supposed to say "Figure ??" at the beginning? or did I miscompile...

DanielSank

02:21

I'm telling you, it needs work :-\

heather

no, no, it's fine, I was just making sure it didn't mess something up

DanielSank

@heather It's either because I haven't added that figure yet, or because some times you have to run twice to get the labels to show up.

The first chapter is kinda rocky at the beginning. I'm working on it. Please don't judge too much yet.

heather

okay, I don't think you've added that figure yet.

I'm not judging at all!

DanielSank

But if you have suggestions, please file issues on the git repo.

heather

it looks good

DanielSank

02:22

Please please please please.

heather

okay

DanielSank

My whole plan for success is based around the idea of actually using feedback from readers to make it good.

I don't want to write yet another crappy book.

heather

another?

I did file an issue; it's really rather minor.

DanielSank

@heather In addition to all the other ones in the world.

@heather Thank you thank you thank you.

heather

@DanielSank oh, okay.

@DanielSank I just noticed one thing: one of the figures seems to be a page and a half below where it is referenced; is that okay, or...?

DanielSank

02:25

@heather It's a real pain to get LaTeX to put figures where you want them.

Don't bother filing issues about that.

heather

okay

DanielSank

I'll do it when the text is all done.

I'm heading home. Might be on later.

Ciao.

heather

it is really nice to use the package float and then do \begin{figure}[H]; that just sets the image at the exact spot in the text you put it in the code

@DanielSank, adios

=)

thanks again for your help

Skyler

hey heather

so lets do a quick recap of everything you've learned today

so feel free to tell me what you felt like were the most important things you wanted to learn today, and what Dan got to cover @heather

and if any questions start popping up just leave them here since that way you can find them again easily

heather

@Skyler, well, we went over bra-ket notation, and how that relates to matrix notation, which really cleared a whole lot of things up.

Skyler

02:37

@heather what questions did it clear up in particular

heather

We also went over linear transformations/matrices and the difference between the two
and then we also went over how in normal algebra, the equation is not dependent on base, which was really cool
we also went over the general format of physics problems

a bit of eigenvalues eigenvectors

@Skyler well, what in the universe it meant, in the first place

I had no clue what to do with it in general contexts.

And how it is like the abstracted version of normal matrix notation, that made it clear how it was different from normal matrix notation.

how to manipulate it, as well - I didn't know that $\langle$something$|$something$\rangle$ meant the dot product of the somethings

he also explained in which order to do differential equations, vector calculus, and multivariable calculus (in the reverse order that I listed it), which was unrelated to linear algebra, but good to know =)

and then he walked me through how matrix multiplication is defined, which was the bulk of our conversation

Skyler

@heather this detail is really helpful in quantum mechanics because later on we start caring about things in quantum mechanics where its not exactly solvable by differential equations

heather

@Skyler, right, and as I'd like to learn QM in some point, with the mathematics and everything, that's going to be really good to know =)

and finally we talked about his book-in-progress

Skyler

(there are only like 10 problems that we have REALLY solved in a neat closed form without sums)

in QM

so a lot of QM is spent trying to describe problems we cant exactly solve in terms of their projections (dot products) to things we have solved

heather

hmm...that's interesting

I'm sorry @Skyler, I'm going to have to go for the night =/

Skyler

02:46

np, feel free to leave any questions you have, before you go can I introduce an idea for a little game using bras and keets

kets*

heather

okay

Skyler

just put something "inside" a ket-bracket, that you can visibly count

like bananas

and then put something like an adjective or specific thing like <colored brown|

and then do <colored brown|banana>

its kinda like doing a dot product, but with countable things

really helpful later on in QM

heather

okay, i'll keep that in mind

Skyler

since the dot product is just a type of projection you do

good night heather

heather

i'll have to look up projection later

good night @Skyler

thanks again

=)

Skyler

02:50

yw, =)

good night

Transcript for

♦ Linear algebra happy fun time with Da…