Looking Glass Universe
What is the Fourier Transform?
updated
This is Karpathy's tutorial: youtu.be/l8pRSuU81PU?si=IZKoAl-YpqEn9Tyj
The code: github.com/karpathy/build-nanogpt/blob/master/train_gpt2.py
My write up as I went: colab.research.google.com/drive/1awhFM8oIGMVTQII-S7rsQGFeJDDxFoEV?usp=sharing
The problems this year: imo-official.org/year_info.aspx?year=2024
Here’s a video solution for question 6: youtube.com/watch?v=7h3gJfWnDoc
AlphaFold 2: youtu.be/3gSy_yN9YBo
AlphaFold 3: youtu.be/CYncNBMPLLk
Here are the notes I was writing in the video: lookingglassuniverse.substack.com/p/deep-dive-into-alphafold-3
Resources:
fast.ai course on diffusion: course.fast.ai/Lessons/part2.html
Algorithmic Simplicity's video on diffusion: youtube.com/watch?v=zc5NTeJbk-k
Computerphile video on diffusion: youtube.com/watch?v=1CIpzeNxIhU
Super helpful video on diffusion transformers: youtube.com/watch?v=fWUwDEi1qlA
Link to the Colab (make a copy of it to fill it in): colab.research.google.com/drive/10TjJAOO-oCGIWEScYF7HTbGInVqJm3yN?usp=sharing
GPT theory videos:
3blue1brown’s super helpful videos on this: youtu.be/eMlx5fFNoYc?si=_7XgY5nOpVaRdThU and youtu.be/wjZofJX0v4M?si=Cc4QkmzN6rsJZuOc
Algorithmic Simplicity’s video give great intuition: youtu.be/kWLed8o5M2Y?si=2WqR04FblltNxbF-
PyTorch nn.Module tutorial: pytorch.org/tutorials/beginner/basics/buildmodel_tutorial.html
When I didn’t understand things, I mostly just asked ChatGPT to explain it to me. The 4o model has just become freely available, so give it a go!
Previous AlphaFold video: youtu.be/3gSy_yN9YBo
The research I referenced that OpenAI did on multimodal models is here: openai.com/index/clip
Claude: claude.ai
Gemini: gemini.google.com
OpenFold: github.com/aqlaboratory/openfold
The original paper about transformers (the technology behind GPT), Attention Is All You Need: arxiv.org/abs/1706.03762
*Helpful videos and blogposts about attention and AlphaFold*
3blue1brown's video on transformers: youtu.be/eMlx5fFNoYc?si=V5k3r6sc5NSMhFt0
Nazim Bouatta's talk on AlphaFold: youtu.be/ri39B0Voujc?si=JpFOlxrVIm0wfmsZ
Simon Kohl's talk on AlphaFold, (he is one of the coauthor's of the algorithm): youtu.be/tTN0MM2CQLU?si=x7I5muFg0lm0VDy3
A very thorough blogpost about the algorithm from UV-Bio: https://www.uvio.bio/alphafold-architecture/
The Openfold github + paper: github.com/aqlaboratory/openfold + biorxiv.org/content/10.1101/2022.11.20.517210v2.full.pdf
Animations by the extremely talented Kathy Sarpi: kathysarpi.com
Link to the full series: youtube.com/playlist?list=PLg-OiIIbfPj3mDFx5zjVPtgiGwZMM4Erw
Links to other photoelectric effect videos:
This one has a very simple at home set up: youtube.com/watch?v=A9SSfZBMaH8
For this one you'd need to buy the device he uses, but the video is so well explained: youtube.com/watch?v=oYnp0WZDhYQ
Here's an alternate way to do the experiment with an aluminium can: youtube.com/watch?v=WO38qVDGgqw
There's so many other good ones! If you see one you think I should add to this list, please leave the name of the video and the channel in a comment!
Huygen's optics' video about photons (highly recommended): youtu.be/SDtAh9IwG-I?si=8kWz13_ZEFnfAu-Y
Confining the light causes it to have more colours. This is explained well in an excellent video by Ben Miles. But at first glance, the experiment seems to be a totally different one than what I explained in the video- so I'll explain what the connection is. In particular, it's only the "single slit" version of the experiment that's relevant for us. I said that the researchers confined a laser to a small space. The technique they used to do this was to have two lasers- one which is the source, and the other which is used to turn on and off a "switch" of sorts. What the switch does is it makes the material in the experiment go from transparent to reflective very quickly, then back. The source laser is shining continuously at the material. But the idea is that for the short while that the material is reflective a little section of the laser beam is reflected. That's the "confined" light- they took a laser beam that's always on and constant and isolated a small section, confining the whereabouts of the light. They then measured the colours of that light and find it's spread out. (This result is at the 8 minute mark)
youtube.com/watch?v=NsVcVW9GI60 especially from 7:07
Part 1 is here: youtu.be/muoIG732fQA?si=_vFy9siMqkOdO1xV
f you want to do this experiment at home, you can! It's very simple.
All you'll need is:
- a weak red laser pointer (the type in cat toys are generally safe)
- polarizing film or polarizing filter. If you have polaroid glasses or certain camera ND filters you may already have this. Otherwise it's available on amazon
- half waveplate (the plastic thing) is this one: edmundoptics.co.uk/f/polymer-retarder-film/14827 (λ/2 Retarder Film (WP280))
- You don't need calcite, but if you want to play with it, you can find it on etsy usually. Look for a sample that's exceptionally clear
Thank you Kathy for the beautiful animations! kathysarpi.com
Part 2: youtu.be/tHfGucHtLqo
If you want to do this experiment at home, you can! It's very simple.
All you'll need is:
- a weak red laser pointer (the type in cat toys are generally safe)
- polarizing film or polarizing filter. If you have polaroid glasses or certain camera ND filters you may already have this. Otherwise it's available on amazon
- half waveplate (the plastic thing) is this one: edmundoptics.co.uk/f/polymer-retarder-film/14827 (λ/2 Retarder Film (WP280))
- You don't need calcite, but if you want to play with it, you can find it on etsy usually. Look for a sample that's exceptionally clear
Links to the other videos mentioned:
Part 1 of this story: youtu.be/yP1kKN3ghOU
3Blue1Brown's explanation of Feynman's proof: youtube.com/watch?v=KTzGBJPuJwM
Experiment:
If you’d like to try the experiment I did at home then you’ll need a phone with Lidar or you can get a laser meter quite cheaply at a hardware store. If you use an iphone, the app I found most reliable for the measurement was this one: apps.apple.com/us/app/lidar-measuring/id1535032210
Code:
The 3D simulation is here: github.com/mithuna-y/speed_of_light_in_a_medium/tree/main/multiple_layers
References:
The Feynman lectures- “Ch 31: On the origin of refractive index” and “Ch 48: Beats”
3Blue1Brown's video: youtube.com/watch?v=KTzGBJPuJwM
Part 2: youtu.be/uo3ds0FVpXs?si=zW8nyuZpn8NshfqC
Experiment:
If you’d like to try this experiment at home then you’ll need a phone with Lidar or you can get a laser meter quite cheaply. The app I found most reliable for the measurement was this one: apps.apple.com/us/app/lidar-measuring/id1535032210
Code:
The 3D simulation is here: github.com/mithuna-y/speed_of_light_in_a_medium/tree/main/multiple_layers
References:
The Feynman lectures- “Ch 31: On the origin of refractive index” and “Ch 48: Beats”
Matter and Interactions 3rd Edition - The quote is from section 24-4 on page 1001. It’s a long passage but I tried to paraphrase it accurately. Here’s the full quote:
How might we measure the speed of propagation of an electromagnetic wave?
One can think of two different approaches:
(a) Follow a wave crest: If you watch one particular wave crest, you will see
that it travels a distance λ (one wavelength) in a time T (one period).
Therefore the speed of the crest is
v = λ / T
(b) Time the arrival of a radiative electric field: One could imagine a different way of measuring the speed of an electromagnetic wave. Suppose that you and a friend synchronize your clocks, then travel to locations that are a distance d apart. Your friend aims a laser at your location, and precisely at time t1, turns on the laser. You record the time t2 at which you first detect the radiative electric field. In the laser light, and knowing the distance between the locations and the elapsed time Δt = t2 - t1, you calculate the speed at which the laser light traveled toward you:
v = d / Δt
In a vacuum, these two ways of measuring the speed of a sinusoidal electromagnetic wave will give the same answer: 3 x 10^8 m/s. However, this will not necessarily be the case if part or all of the space through which the light wave travels is filled with a medium such as water, glass, or even air. In this case, method 2 (measuring the time required for information about a change in the electromagnetic field to travel a given distance) will still give 3 x 10^8 m/s. However, method 1 (timing the interval between crests in a steady state electromagnetic wave inside the medium) will give a different answer, which will almost always be less than 3 x 10^8 m/s.
youtu.be/v_uBaBuarEM
Safety and where to get the supplies:
I would love for you to try this experiment but please be careful with the lasers. If you're going to buy a green laser then it's crucial that you buy a proper one. Green lasers emit light a lot of invisible IR light and in cheap lasers this often isn't filtered out well. Blue/ violet lasers can also be dangerous for another reason. Our eyes are terrible at seeing these wavelengths, so the laser will look much less strong than it is, which means you might be playing with a dangerously strong laser without knowing it. Red lasers are generally the safer bet. I saw the effect I was looking for even when I used cheap ~1mW red lasers, so it will still work for you!
If you'd like to invest in a nice laser though, this article has some trustworthy green laser brands: planetguide.net/astronomy-laser-pointer
Another way to buy lasers that are eyesafe is to get them from school science lab suppliers.
(Note: in some countries it's illegal to sell lasers over 1mW. Please check the laws where you live. In the USA the limit is 5mW)
The smoke machine I got used glycerol. I’m very suspicious of breathing in that smoke though, especially since the room can't be too well ventilated or it won't work. Fine particulate matter is a serious health risk in general so even though glycerol is nontoxic I think it may still be hazardous to inhale. I recommend wearing an airtight n95 mask or similar while doing this experiment.
It found it a bit tricky to source the double slit cheaply in Australia- your best bet might be a science lab supplier. In the USA you can get it on amazon though.
Video credits:
The beautiful animations in this video are made by Kathy Sarpi: kathysarpi.com
Thank you to Screen Australia and Google Australia for funding this project, and to the wonderful people at Screen Australia who helped me throughout the process.
Thanks also to all my beta testers (aka friends)!!
Btw, just because I think people will be confused, my field (quantum computing) is cross disciplinary, but at Cambridge it was in the maths department. I never know whether to call myself a physicist or a mathematician.
Here are a few papers that prove this result (there are many more though!):
David Deutsch arxiv.org/abs/quant-ph/9906015
Sabine Hossenfelder (yes that one) arxiv.org/abs/2006.14175
Wojciech Zurek arxiv.org/pdf/quant-ph/0101012
Lucien Hardy arxiv.org/pdf/quant-ph/0101012
0:00 - 1:38 Intro to the Born Rule
1:38 - 3:14 Equal probability case
3:14- 3:51 Nerd stuff
3:51- 5:00 Finishing the equal probability case
5:00- 6:00 Talking about my former employer
6:00- 10:14 Example with unequal probabilities
10:14- 12:16 A more general example
12:16- 14:04 How do you generalise?
Music:
Spinning Monkeys by Kevin MacLeod incompetech.com/music/royalty-free/music.html
Why don't I experience being in two worlds? youtu.be/4dCrNMqvYyg
The many worlds interpretation of quantum mechanics is often criticised for being excessive. Isn't it crazy that there are an infinite number of worlds splitting all the time? In this video I give an introduction to what many worlds actually is, and why I think it's actually a simplification of standard quantum mechanics.
Book recommendations:
The best book on MW in general and the one that inspired this video is Emergent Multiverse by Wallace.
Another fantastic book is Decoherence: and the quantum-to-classical transition by Schlosshauer. It isn’t specifically about MW, and is useful to understand decoherence in general.
Here’s my advice as a person who isn’t a doctor or an expert but has had some bad experiences with this condition.
1. It can be difficult to diagnose this condition partly because the symptoms are so different for everyone and partly because it’s hard to know what counts as “severe” pain. I thought the amount of pain I was having was normal for many years. If you have periods that are painful enough to be bothering you then it’s worth talking to a doctor.
2. GO TO A DOCTOR WHO SPECIALISES IN ENDO (once you’ve been diagnosed). Please please please don’t just go to a normal gynecologist. I did that and it was a huge mistake that cost thousands of dollars and caused damage that couldn’t even be fixed in my second operation. The surgery for endo is quite complicated and requires dedicated training to master, and yet the majority of doctors offering it haven’t done that training. For more details on how to find someone good I recommend that you:
4. Read some books about endo. In particular, I liked How to Endo by Bridget Hustwaite. It was approachable and full of useful advice. When I first got diagnosed I looked up endo online but that was pretty useless. It was only after reading this book that I got some idea of what was going on. That said, this condition is not well understood yet and everyone has their own methods of treating it that might not work for you, so don’t trust everything you read everywhere.
Here’s the diagnosis app I used (this isn’t an endorsement): ada.com
If you want to learn about causal Bayesian Networks, The Book of Why by Mackenzie and Pearl is excellent.
Timestamps:
00:00 My medical story
02:48 Using Bayes' Rule for diagnosis
06:17 Causal Bayesian networks
12:34 Homework
Here are a few citations:
2:20 I claimed that entanglement has been shown to cause a small (polynomial time) improvement in quantum computing in a particular circumstance. That was shown in an amazing paper that used Bell inequalities to prove you the fact: arxiv.org/abs/1704.00690
2:45 On the role of entanglement in the quantum-computational speed up: arxiv.org/abs/quant-ph/0201143
Chapter 4 of my PhD covers the entanglement project. This is the most up to date source on it- we haven't uploaded the new version to the arXiv yet: repository.cam.ac.uk/handle/1810/315974
youtube.com/playlist?list=PLg-OiIIbfPj3vwCUeRWweRcnZMWJNXb27
0:00 Intro
0:43 Do something else first
3:11 Look for the right things in a supervisor
4:18 Choose a university with a lot happening
7:09 ...maybe don't do a PhD in the US
8:36 Final words of discouragement
This video used this delightfully funny track: freemusicarchive.org/music/Kevin_MacLeod
I also highly recommend this site for understanding what the data we have: ourworldindata.org/coronavirus
And this video from 3Blue1Brown that explains the mathematics of exponential growth:
youtube.com/watch?v=Kas0tIxDvrg
REFERENCES:
0:58 To find the fatality rates for different age groups, see: ourworldindata.org/coronavirus
1:05 ‘You are likely to spread it to 2-3 other people’ claim is based on the paper: github.com/cmrivers/ncov/blob/master/COVID-19.pdf
1:40 Death rates for underlying conditions quoted were based on this information: ourworldindata.org/coronavirus#deaths-from-covid-19
2:51 ‘Number of official cases in the USA is likely to be a huge underestimate’ because of how few test have been administered. See Chart 8 on this page: vox.com/future-perfect/2020/3/12/21172040/coronavirus-covid-19-virus-charts
3:12 The statement “without drastic measures, the problem gets 10 times worse in a week” is based on the data in the graph below. The data from this graph comes from WHO’s official count. The slope of the lines in this graph determine how quick the so-called ‘doubling time’ is: the time till the problem is 2 times worse. You can see that initially, before much has been done, this slope is often consistent with a doubling time of less than 2 days, but it goes to between 2 and 3 days once a country is at least doing something, but has not imposed a lockdown on nonessential gatherings and closed schools etc. The USA (before the recommendation that just came out) had a doubling time of around 2 days (although it’s hard to say because of the testing situation mentioned below). That translates to the problem being 10 times worse in just under a week. In other places the doubling time was more like 2.5 days, in which case it’s 10 times worse in 8 days. Public awareness and bans on mass gatherings seem to help to bring the doubling time, usually to around 4 days. But when lockdowns are imposed it seems to get dramatically lower.
ourworldindata.org/grapher/covid-confirmed-cases-since-100th-case
4:34 Flatten the curve infographic: twitter.com/SiouxsieW/status/1236721200291655680
5:20 ‘350 thousand beds’ claim can be computed from these two sources:
data.oecd.org/healtheqt/hospital-beds.htm
statista.com/statistics/185904/hospital-occupancy-rate-in-the-us-since-2001
5:32 ‘3-5% death rate in overwhelmed healthcare systems, and potentially much higher’ claim is from source:
wwwnc.cdc.gov/eid/article/26/6/20-0233_article
6:00 The Washington Post simulation is available here: washingtonpost.com/graphics/2020/world/corona-simulator
As well as Kevin Simler’s playable simulation (highly recommended!): meltingasphalt.com/outbreak
8:09 Virus symptoms infographic: ourworldindata.org/coronavirus#
8:24 ‘20%+ of infections from those who have no symptoms is based on this paper: cmmid.github.io/topics/covid19/control-measures/pre-symptomatic-transmission.html
ONLINE RESOURCES FOR LEARNING (to keep you occupied):
www.brilliant.org: You can signup without putting in your credit card details to do the daily challenges. You can also try the first chapter of every one of their courses for free this way. At the moment they have a free trial that let’s you do everything on the site, but you will need a credit card for that one. Disclaimer, I’m going to start working for Brilliant soon, but they’ve not asked me to do this or anything- I just really like the site.
3blue1brown has a lot of fantastic videos and I’m sure you’ve seen them, but have you watched his whole linear algebra series? It’s a gem, and useful to know too!: youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
Socratica has beautiful in depth videos on all sorts of topics. I’m going to take this chance to follow their python course: youtube.com/channel/UCW6TXMZ5Pq6yL6_k5NZ2e0Q
Science4All is run by the extremely likable Frenchman Dr. Lê Nguyên Hoang. Why not learn some general relativity? I know you’ve always wanted to:
youtube.com/watch?v=0SUs_o6EEVs&list=PL8ovs-QtxcNydaiz8LSPpo8G0fatCqdTQ
Leonard Susskind's Theoretical Minimum series: youtube.com/playlist?list=PL701CD168D02FF56F
ACKNOWLEDGEMENTS: Thank you Le from Science4All for prompting me to make this video by pointing out how urgent this message is, and helping so much with the script. Thank you also to Josh Silverman for his many invaluable comments on the script that made this a much better video. Thank you Dr Rohin from MedLifeCrisis for offering to check the script even though he’s working so hard because of COVID19. And thank you to Dr M.K. for vetting the medical accuracy here. There are no doubt mistakes left, but those are all my own.
There's a lot more info in the blog post! lookingglassuniver.wixsite.com/blog
Reddit: old.reddit.com/r/Looking_glass_u
Scott's book about learning:
scotthyoung.com/blog/ultralearning
About me: in case you’re wondering, my name is Mithuna Yoganathan and I’m currently a PhD student studying theoretical physics at the University of Cambridge. If you want to know more about me, I made a video on my path into physics: youtu.be/nEhhQtq9gp4
And more info about GiveWell, including how they do their assessments, is here: givewell.org/how-we-work/process
Many people have the view that charity is either ineffective or corrupt. In this video we discuss why it certainly doesn't have to be that way -if we take a more scientific approach to doing good. (Just to be clear though, this isn't meant to disparage other charities and/or Project for Awesome. I personally am a fan of Partners in Health (one of the main P4A charities) as well as many others that GiveWell doesn't currently list as it's top charities.)
(Ad revenue from this video will go to GiveWell, by the way)
Reading recommendation:
If you enjoyed this video and want to know more, I really recommend Doing Good Better, by William Macaskill. I don't fully agree with every aspect of it (in particular, it takes a very utilitarian approach, even though you can take other moral views and come to similar conclusions). But it is a really excellent introduction to this topic, which is called Effective Altruism.
Citations:
My account of Michael Kremer's work comes from Doing Good Better, but the original papers are below. Just to be clear though, this doesn't mean that textbooks, flipcharts and more teachers are never useful. Just that they were not in these specific places and times tested.
General overview: "Randomized Evaluations of Educational Programs in Developing Countries: Some Lessons" www.jstor.org/stable/3132208?
Textbooks: "Many Children Left Behind?
Textbooks and Test Scores in Kenya" www.povertyactionlab.org/evaluation/textbooks-and-test-scores-kenya
Flipcharts: "Retrospective vs. prospective analyses of school inputs: the case of flip charts in Kenya" www.poverty-action.org/study/flipcharts-and-school-inputs-kenya
Price of deworming tablets: www.evidenceaction.org/dewormtheworld
The study I cited with the 108 health interventions:
Jamison, Dean, et al. (eds.). 2006. Disease Control Priorities in Developing Countries (second
edition) Oxford University Press
But the results are discussed in a paper available online (it's a good read!): cgdev.org/sites/default/files/1427016_file_moral_imperative_cost_effectiveness.pdf
FINALLY, if you've got this far, I think you'll really enjoy this: 80000hours.org
Vectors and Bases: youtu.be/3ZfrJ0Sk5iY
Matrices and Linear Transforms: youtu.be/CBIO4xJ1Cok
Matrix inverses: youtu.be/ESKcF8XFzLM
This video is about changing the basis (or coordinate system) of a matrix or a vector. While the change of basis formula is often presented as something to just memorise, we'll see it's actually very very straightforward to understand.
Hints for homework:
1. You can do it! Think about what it does to the basis vectors!
2. Remember, in the original case you first translate the input vector from Bob's basis to Alice's so you can apply M_a, then translate back. A similar idea applies, but you need to translate to something M_a can take as input.
Answers to homework below!!!
Answers
The answer to question one is 2.
The answer to question two is 3.
Link to my linear transformation/ matrix video: youtu.be/CBIO4xJ1Cok
Link to my vectors and bases video: youtu.be/3ZfrJ0Sk5iY
Link to my (unlisted) original version of this video: youtu.be/pLz_ln0ByXo
This video is about matrix inverses, and in particular, I try to give a bit more intuition for them- rather than just giving you the formula for the determinant, Cramner’s rule, the inverse of 2x2 and 3x3 matrices etc. Along the way, we cover some topics that don’t receive enough attention in linear algebra (at least, they didn’t in my math classes), like the left inverse, and why non square matrices don’t have an inverse. Finally, we will learn an intuitive condition for when the inverse exists.
Homework questions
1. What is the inverse of the following matrix:
1 2
2 5
a)
-1 -2
-2 -5
b)
1 1/2
1/2 1/5
c)
1 -2
-2 5
d)
5 -2
-2 1
2. Prove that, for a square matrix, the left inverse = the inverse
3. Prove the above for the following matrix:
1 2
2 5
by first doing these 2 questions
a) Show that M(1, 0) and M(0,1) for a basis. I.e. the old basis gets mapped to a new basis.
b) Show MLv = v for any vector (this is enough to show the M undoes L, and hence show that the left inverse is equal to the inverse).
To do this, first write v as a linear combo of M(1, 0) and M(0,1).
Hints:
3.
a) is a straightforward if you watch my first linear algebra video: youtu.be/3ZfrJ0Sk5iY
b) Sub the expression for v into MLv. Then use linearity!
Q2.
a) (Proving that the basis is always mapped to another basis) This is a bit tricky. I've written out the solution but stop reading at the point you think you know what to do and try it.
Assume v1,...,vn are a basis. Apply M to them to get Mv1,..., Mvn.
They're a basis if they're still linearly independent. But if they were, say:
Mv1= a Mv2+... + d Mvn
Apply L to both sides:
LMv1= L(a Mv2+... + d Mvn)
= a LMv2+... + d LMvn
using LM=identity:
v1= a v2+ ...+ d vn.
But this must be false, since v1,..., vn are linearly indep.
To see it done (SPOILER ALERT) 9:20 in youtu.be/pLz_ln0ByXo
b) Once you have that, proceed in a very similar way to Q3, part b)
Previous video on vectors and bases (watch this first): youtube.com/playlist?list=PLg-OiIIbfPj3Wldtb0QfV0Yse8tL2nLGm
Next video:
youtu.be/ESKcF8XFzLM
Matrices are often presented as a useful bookkeeping/ commutation tools to students- but there’s much more to them. When you understand what a Matrix really is so many parts of Linear Algebra will be completely obvious to you… including the formula for matrix multiplication and the fact that matrices don’t commute. So here's the big secret: A matrix is a linear transformation that eats a vectors and outputs another vector.
Homework questions:
Not all sized matrices can be multiplied together. Think about it in terms of them representing transformations from one space to another, and figure out which size matrices can be multiplied and explain why in the comments.
Consider a transformation that takes a 3d vector, and adds some fixed vector k to it. Say k is the vector 7 3 3. Is this a linear transformation or not? brilliant.org/practice/linear-transformations/?p=1
Imagine you have a matrix A that multiplies the first basis vector by 2, and the second basis vector by 6. How do you write A in this basis? brilliant.org/practice/linear-transformations/?p=3
Music: Epidemic sound, Summer nights 2
This video is an Introduction to Matrices but could be useful revision for school/university. If you have an exam, good luck!
Next video: youtu.be/CBIO4xJ1Cok
Check out Brilliant: brilliant.org/LookingGlassUniverse
DON'T FORGET TO DO YOUR HOMEWORK:
Prove these 2 statements about bases
1. If you have two different bases for the same space, then they must have the same number of basis elements in them. (E.g, there are many different choices of basis for the plane, but no matter what basis you choose, there are only 2 vectors in each basis.)
2. Once you pick a basis (say {v_1, v_2}), there's only one correct way to write another vector as a linear combination of the basis vectors. Eg, say v=a v_1+ b v_2. Then you can't also write v=a' v_1+ b' v_2, where a' and b' are different from a and b.
The multiple choice questions from Brilliant:
Q1. Which of these vectors is redundant (i.e. can be written as a linear combination of the other 2):
i) (1 2 3)
ii) (1 3 5)
iii) (2 5 8)
iv) Each of the above
The full solution is available here: brilliant.org/practice/linear-independence/?p=2
Q2. Consider the following 3 vector spaces:
A= The vector space spanned by {(1 2)}
B= The vector space spanned by {(1 2), (2 3)}
C= The vector space spanned by {(1 2), (2 4), (3 6)}
Question: which of the following is true?
i) A is a subspace of B, which is a subspace of C
ii) C is a subspace of A, which is a subspace of B
iii) B is a subspace of C, which is a subspace of A
iv) A is a subspace of B and C, which are not subspaces of each other
The full solution is here: brilliant.org/practice/subspaces-and-span/?p=6
ANSWERS FOR THE BRILLIANT.ORG QUESTIONS:
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Q1) D
Q2) ii
HINTS FOR PROOF QUESTIONS:
2 is easier, so let's do that first
Hint 2.1 Let's do the case with just 2 basis vectors first. If there are 2 basis vectors v_1 and v_2, the one thing you know about them is that they are not just a multiple of each other (otherwise it wouldn't be a basis). Try and get a contradiction with this fact.
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Hint 2.2 Assume v= a v_1+ b v_2 = a' v_1+ b' v_2, but a and a' aren't equal, and b and b' aren't equal.
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Hint 2.3 Use the above equation to write a relationship between v_1 and v_2. Oh no, that looks like they are multiples of each other!
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Hint 2.4 Now do the case where there are n basis vectors. What you know about them is that you can't write them as linear combinations of the others. Try and get a contradiction with this fact.
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Hint 1.1. Imagine you had 3 vectors and they span 2D space. Doesn't that one of them is redundant? The following in this case first:
B1={u_1,u_2}
B2={v_1, v_2,v_3}
Write each u_i in terms of B1.
Remember that there is a redundancy in B2 if you can write c u_1+ d u_2= u_3. So write this, and let's see if we can find a solution for c and d.
Plug in your equations for u_i into c u_1+ d u_2= u_3
You now have a vector on the right hand side in B1 and a vector on the left hand side in B1
Using the result from question 2 (dammit, I really should have swapped the order of these questions), you know that the coefficient in front of v_1 and v_2 must be the same on both sides (since there is only one unique way to write a vector in B1)
So now you have 2 linear equations with 2 unknowns (c and d- everything else is 'known').
Show that they only don't have a solution for c and d if B2 was actually linearly dependent all along. (Yes this will require you to know some linear algebra to do efficiently (although technically possible without)).
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When you assume that the two bases can be any size it's most efficient to do this with linear algebra (sorry!!)
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The solution: yutsumura.com/if-there-are-more-vectors-than-a-spanning-set-then-vectors-are-linearly-dependent
Music: Epidemic sound, Summer nights 2
In particular, I talked a lot about the measurement problem in this video.
Thanks to Vlogbrothers for their sponsorship of this video! I really really appreciate it!
If you'd like to help transcribe this video into another language, that would be amazing: http://www.youtube.com/timedtext_vide...
Homework questions:
1. We talked about doing the double slit experiment with small things like electrons. But is there any reason in theory we can't do it with bigger things? If not, what would you need to do to make it work?
2.We didn't measure where the electron went in the middle of the experiment. But it is possible. If we measured each door to see if the electron went through it, what would we find? And what would happen to the pattern on the back wall?
3. There is another way to interpret the double slit experiment, which I think is really cool- it's called Pilot wave theory. I made a video about it. In that theory, the particle does go through just one door, even though our argument said that's not possible. Find the flaw in our argument. Also comment on why particles still act strange in Pilot wave theory.
Music: Thanks Falcxne for allowing me to use this song: https://m.soundcloud.com/falcxne/phas...
THE NEW SERIES HAS STARTED: youtu.be/8Dso6Fv1FUw
Here are the links to videos I've now made unlisted (this means that anyone with the link can view them, but they won't turn up in Youtube's search anymore).
The old old (!) videos on Quantum mechanics (from 5+ years ago):
The wave particle duality: youtu.be/zDQH5x7svfg
Superposition ft. Schrodinger's Cat: youtu.be/zDQH5x7svfg
What is a measurement?: youtu.be/KujEMGZGUTI
Quantum Eraser: youtu.be/XcZ3jI1Ph7A
Quantum Eraser explained: youtu.be/sQfSm6o-KlQ
Delayed Choice Quantum Eraser: youtu.be/MW-AemjSVGY
Heisenberg Uncertainty Principle: youtu.be/ZpwZgOumTrs
New old series (from 3 years ago):
Introduction to quantum mechanics: youtu.be/b_ddt6J1Bio
The Wave Function: youtu.be/02eZMf17wFs
(I will unlist some of the others later as I upload more in this new series.)
Links to videos referenced:
Veritasium's video: youtu.be/WIyTZDHuarQ
My old Bohmian mechanics video: youtu.be/rbRVnC92sMs
Contextuality: youtu.be/Qz4CHI_W-TA
Entanglement and the EPR paradox: youtu.be/Xzmp7byh77E
Also see:
PBS spacetime's excellent video: youtu.be/RlXdsyctD50
This amazing video about 'surreal paths' in Bohmian mechanics (this channel is also very worth checking out): youtu.be/CCW93koLNYY
Homework:
Prove that faster than light communication doesn't work for the state in the video, when Alice measures up and down-ness, but Bob measures left-rightness.
Look up and then explain the usefulness of a one-time pad. Also explain whether a one time pad coming from a bunch of shared particles is secure if Eve is trying to measure Bob's particle's before he does.
The EPR paradox, that we met in a previous video, tells us 2 entangled particles can effect each other no matter now far away they are. But then why can't we use them to send instant messages across the universe? Einstein's relativity tells us it would be a disaster if we could!
Homework:
Prove that faster than light communication doesn't work for the state in the video, when Alice measures up and down-ness, but Bob measures left-rightness.
Look up and then explain the usefulness of a one-time pad. Also explain whether a one time pad coming from a bunch of shared particles is secure if Eve is trying to measure Bob's particle's before he does.
Solution to homework question 1:
Sorry Youtube doesn't allow certain brackets in the description, so I've put the solution as a comment.