Its possible to prove the Born Rule of quantum mechanics Looking Glass Universe 2022-08-11 | If you'd like to try Brilliant: brilliant.org/LookingGlassUniverseHere are a few papers that prove this result (there are many more though!):David Deutsch arxiv.org/abs/quant-ph/9906015Sabine Hossenfelder (yes that one) arxiv.org/abs/2006.14175Wojciech Zurek arxiv.org/pdf/quant-ph/0101012Lucien Hardy arxiv.org/pdf/quant-ph/01010120:00 - 1:38 Intro to the Born Rule1:38 - 3:14 Equal probability case3:14- 3:51 Nerd stuff3:51- 5:00 Finishing the equal probability case5:00- 6:00 Talking about my former employer6:00- 10:14 Example with unequal probabilities10:14- 12:16 A more general example12:16- 14:04 How do you generalise?Music:Spinning Monkeys by Kevin MacLeod incompetech.com/music/royalty-free/music.html
How I used AI to understand a huge codebase Looking Glass Universe 2024-05-09 | ChatGPT has a fairly small limit on the size of files you can upload to it. Claude has a much larger limit, which makes it very helpful for uploading a big repo. In this example I used the paid versions of both (because they’re much better). However, Claude is blocked in a lot of places right now. If that's the case for you, try using Google's Gemini instead. It's context window is even larger, but unfortunately you can't upload docs to it right now. So instead you'll need to copy and paste the entire thing in... Painful, but it will work. For this you will probably still need to use the paid version to get the entire codebase in.Claude: claude.aiGemini: gemini.google.comOpenFold: github.com/aqlaboratory/openfold
How AlphaFold solves protein folding Looking Glass Universe 2024-04-26 | AlphaFold paper: nature.com/articles/s41586-021-03819-2The 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=V5k3r6sc5NSMhFt0Nazim Bouatta's talk on AlphaFold: youtu.be/ri39B0Voujc?si=JpFOlxrVIm0wfmsZSimon Kohl's talk on AlphaFold, (he is one of the coauthor's of the algorithm): youtu.be/tTN0MM2CQLU?si=x7I5muFg0lm0VDy3A 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
What *is* a photon? Looking Glass Universe 2024-02-13 | This video was supported by Screen Australia and Google through the Skip Ahead initiative.Animations by the extremely talented Kathy Sarpi: kathysarpi.comLink to the full series: youtube.com/playlist?list=PLg-OiIIbfPj3mDFx5zjVPtgiGwZMM4ErwLinks to other photoelectric effect videos:This one has a very simple at home set up: youtube.com/watch?v=A9SSfZBMaH8For this one you'd need to buy the device he uses, but the video is so well explained: youtube.com/watch?v=oYnp0WZDhYQHere's an alternate way to do the experiment with an aluminium can: youtube.com/watch?v=WO38qVDGgqwThere'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-YConfining 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
What can my homemade quantum computer do? Looking Glass Universe 2023-12-22 | This video was supported by Screen Australia and Google through the Skip Ahead initiative.Part 1 is here: youtu.be/muoIG732fQA?si=_vFy9siMqkOdO1xVf 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
I made a (useless) quantum computer at home Looking Glass Universe 2023-12-17 | This video was sponsored by Screen Australia and Google Australia through the SkipAhead Initiative.Thank you Kathy for the beautiful animations! kathysarpi.comPart 2: youtu.be/tHfGucHtLqoIf 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
Phones measure the speed of light ☝️Link to the full video ☝️ Looking Glass Universe 2023-12-14 | ...
I dont know why light slows down in water. (part 2) Looking Glass Universe 2023-11-30 | Supported by Screen Australia and Youtube through the Skip Ahead initiative.Links to the other videos mentioned:Part 1 of this story: youtu.be/yP1kKN3ghOU3Blue1Brown's explanation of Feynman's proof: youtube.com/watch?v=KTzGBJPuJwMExperiment: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/id1535032210Code:The 3D simulation is here: github.com/mithuna-y/speed_of_light_in_a_medium/tree/main/multiple_layersReferences:The Feynman lectures- “Ch 31: On the origin of refractive index” and “Ch 48: Beats”
I didnt believe that light slows down in water (part 1) Looking Glass Universe 2023-11-30 | Supported by Screen Australia and Youtube through the Skip Ahead initiative.3Blue1Brown's video: youtube.com/watch?v=KTzGBJPuJwMPart 2: youtu.be/uo3ds0FVpXs?si=zW8nyuZpn8NshfqCExperiment: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/id1535032210Code:The 3D simulation is here: github.com/mithuna-y/speed_of_light_in_a_medium/tree/main/multiple_layersReferences: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 seethat it travels a distance λ (one wavelength) in a time T (one period).Therefore the speed of the crest isv = 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 / ΔtIn 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.
A surprise double slit pattern Looking Glass Universe 2022-11-13 | Here’s the original video: youtu.be/v_uBaBuarEM
I did the double slit experiment at home Looking Glass Universe 2022-11-11 | Supported by Screen Australia and Youtube through the Skip Ahead initiative. 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-pointerAnother 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.comThank 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)!!
bad at math is a myth Looking Glass Universe 2022-08-15 | I gave a talk at my high school today and it got me thinking again about my experience of being labelled "bad at mathematics" and eventually doing my PhD at the University of Cambridge. 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.
The biggest problem in the Many Worlds theory of quantum mechanics Looking Glass Universe 2022-08-04 | Here's the next video youtu.be/cKlRnutiv-k
How is energy conserved in Many Worlds? Looking Glass Universe 2022-05-07 | If the world splits every time you do a quantum measurement... how is energy conserved??
If there are “Many Worlds why don’t you experience it? Looking Glass Universe 2022-04-30 | Here's my previous video on the many worlds interpretation: youtube.com/watch?v=xBlpOGdk-0U&t=46s
Many Worlds is a simplification of quantum mechanics Looking Glass Universe 2022-04-22 | Follow up videos answering these questions:Why don't I experience being in two worlds? youtu.be/4dCrNMqvYygThe 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.
Making a mathematical mural Looking Glass Universe 2022-03-03 | This video goes into the mathematics behind the "infinite chessboard" fractal that I designed.
This calculation tells you your chances of being sick Looking Glass Universe 2021-08-21 | If you (or a friend) have endo symptoms: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.comIf you want to learn about causal Bayesian Networks, The Book of Why by Mackenzie and Pearl is excellent.Timestamps: 00:00 My medical story02:48 Using Bayes' Rule for diagnosis06:17 Causal Bayesian networks12:34 Homework
Is entanglement the key to quantum computing? Looking Glass Universe 2021-05-07 | In this video I describe my PhD research in quantum computing. Is entanglement really crucial to a quantum computer? The standard wisdom says yes but I wasn't so sure.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/1704.006902:45 On the role of entanglement in the quantum-computational speed up: arxiv.org/abs/quant-ph/0201143Chapter 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
What math research feels like (for me) Looking Glass Universe 2021-03-13 | People usually think math is a dry uncreative subject. It's really not at all though. Doing math research was surprisingly emotional. There's a joy to doing it that's hard to explain.
Should you do a PhD? | PhD in theoretical physics at the University of Cambridge Looking Glass Universe 2021-01-22 | This advice applies most for people looking to do a PhD in the UK in physics/ mathematics, although some of it is more general. Please watch other people's videos on this topic as well to get a broader perspective! youtube.com/playlist?list=PLg-OiIIbfPj3vwCUeRWweRcnZMWJNXb270:00 Intro0:43 Do something else first3:11 Look for the right things in a supervisor4:18 Choose a university with a lot happening7:09 ...maybe don't do a PhD in the US8:36 Final words of discouragement
🎄 Making an overly math-y (ugly) Christmas tree 🎄 Looking Glass Universe 2020-12-24 | This video is just a little fun :) Merry Christmas!This video used this delightfully funny track: freemusicarchive.org/music/Kevin_MacLeod
Coronavirus: why you should stay home Looking Glass Universe 2020-03-18 | Here's an informative video on social distancing by a Dr Rohin from MedLifeCrisis: youtu.be/ofSLpDQx1bAI also highly recommend this site for understanding what the data we have: ourworldindata.org/coronavirusAnd this video from 3Blue1Brown that explains the mathematics of exponential growth:youtube.com/watch?v=Kas0tIxDvrgREFERENCES:0:58 To find the fatality rates for different age groups, see: ourworldindata.org/coronavirus1:05 ‘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.pdf1:40 Death rates for underlying conditions quoted were based on this information: ourworldindata.org/coronavirus#deaths-from-covid-192:51 ‘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-charts3:12 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-case4:34 Flatten the curve infographic: twitter.com/SiouxsieW/status/12367212002916556805:20 ‘350 thousand beds’ claim can be computed from these two sources:data.oecd.org/healtheqt/hospital-beds.htmstatista.com/statistics/185904/hospital-occupancy-rate-in-the-us-since-20015:32 ‘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_article6:00 The Washington Post simulation is available here: washingtonpost.com/graphics/2020/world/corona-simulatorAs well as Kevin Simler’s playable simulation (highly recommended!): meltingasphalt.com/outbreak8:09 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.htmlONLINE 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_abSocratica 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_k5NZ2e0QScience4All 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-QtxcNydaiz8LSPpo8G0fatCqdTQLeonard Susskind's Theoretical Minimum series: youtube.com/playlist?list=PL701CD168D02FF56FACKNOWLEDGEMENTS: 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.
How to learn Quantum Mechanics on your own (a self-study guide) Looking Glass Universe 2019-08-08 | This video gives you a some tips for learning quantum mechanics by yourself, for cheap, even if you don't have a lot of math background. There's a lot more info in the blog post! lookingglassuniver.wixsite.com/blogReddit: old.reddit.com/r/Looking_glass_uScott's book about learning: scotthyoung.com/blog/ultralearningAbout 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
Electrons aren’t actual waves Looking Glass Universe 2019-06-09 | What does "the electron is a wave" actually mean? Be careful not to take the statement too literally.
From being terrible at math to a quantum physicist - my journey Looking Glass Universe 2019-03-05 | My path into theoretical physics wasn't a super traditional, since I started off very very bad at math. I hope this video shows you there's no such think as someone who isn't a maths person- since I ended up majoring in pure maths and loving it.
How to take a scientific approach to charity Looking Glass Universe 2018-12-07 | Vote for GiveWell on the Project for Awesome website!: http://www.projectforawesome.com/watch?v=YMxtImOzyGwAnd more info about GiveWell, including how they do their assessments, is here: givewell.org/how-we-work/processMany 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-kenyaFlipcharts: "Retrospective vs. prospective analyses of school inputs: the case of flip charts in Kenya" www.poverty-action.org/study/flipcharts-and-school-inputs-kenyaPrice of deworming tablets: www.evidenceaction.org/dewormtheworldThe study I cited with the 108 health interventions:Jamison, Dean, et al. (eds.). 2006. Disease Control Priorities in Developing Countries (secondedition) Oxford University PressBut 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.pdfFINALLY, if you've got this far, I think you'll really enjoy this: 80000hours.org
The meaning of the dot product | Linear algebra makes sense Looking Glass Universe 2018-11-11 | This video explains why the dot product is about how much vectors point the same way.
Change of basis explained simply | Linear algebra makes sense Looking Glass Universe 2018-09-13 | This video is part of a linear algebra series:Vectors and Bases: youtu.be/3ZfrJ0Sk5iYMatrices and Linear Transforms: youtu.be/CBIO4xJ1CokMatrix inverses: youtu.be/ESKcF8XFzLMThis 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!!!AnswersThe answer to question one is 2.The answer to question two is 3.
A simple condition for when the matrix inverse exists | Linear algebra makes sense Looking Glass Universe 2018-08-26 | Check out Brilliant.org: brilliant.org/LookingGlassUniverse (it's free to sign up- you don't need to enter credit card details)Link to my linear transformation/ matrix video: youtu.be/CBIO4xJ1CokLink to my vectors and bases video: youtu.be/3ZfrJ0Sk5iYLink to my (unlisted) original version of this video: youtu.be/pLz_ln0ByXoThis 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 questions1. What is the inverse of the following matrix: 1 22 5a)-1 -2-2 -5b)1 1/21/2 1/5c) 1 -2-2 5d)5 -2-2 12. Prove that, for a square matrix, the left inverse = the inverse3. Prove the above for the following matrix:1 22 5by first doing these 2 questionsa) 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/3ZfrJ0Sk5iYb) 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 MvnApply L to both sides:LMv1= L(a Mv2+... + d Mvn)= a LMv2+... + d LMvnusing 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_ln0ByXob) Once you have that, proceed in a very similar way to Q3, part b)
Matrices, matrix multiplication and linear transformations | Linear algebra makes sense Looking Glass Universe 2018-07-20 | Brilliant.org: brilliant.org/LookingGlassUniversePrevious video on vectors and bases (watch this first): youtube.com/playlist?list=PLg-OiIIbfPj3Wldtb0QfV0Yse8tL2nLGmNext video: youtu.be/ESKcF8XFzLMMatrices 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=1Imagine 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=3Music: Epidemic sound, Summer nights 2This video is an Introduction to Matrices but could be useful revision for school/university. If you have an exam, good luck!
Vector addition and basis vectors | Linear algebra makes sense Looking Glass Universe 2018-06-28 | Vectors may seem very difficult when you're first introduced to them, but I hope this video helps you see they're not that scary! This video will be especially useful for vectors in physics. We'll cover vector addition and what vectors are. This is the start of a whole series of linear algebra, and I will cover vectors, adding vectors physics, the scalar product, matrices, eigenvalues/ eigenvectors and Dirac notation.Next video: youtu.be/CBIO4xJ1CokCheck out Brilliant: brilliant.org/LookingGlassUniverseDON'T FORGET TO DO YOUR HOMEWORK:Prove these 2 statements about bases1. 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 aboveThe full solution is available here: brilliant.org/practice/linear-independence/?p=2Q2. 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 Cii) C is a subspace of A, which is a subspace of Biii) B is a subspace of C, which is a subspace of Aiv) A is a subspace of B and C, which are not subspaces of each otherThe full solution is here: brilliant.org/practice/subspaces-and-span/?p=6ANSWERS FOR THE BRILLIANT.ORG QUESTIONS:********Q1) DQ2) iiHINTS FOR PROOF QUESTIONS:2 is easier, so let's do that firstHint 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. ***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. ***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!***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. ***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_3You now have a vector on the right hand side in B1 and a vector on the left hand side in B1Using 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)). ***When you assume that the two bases can be any size it's most efficient to do this with linear algebra (sorry!!)***The solution: yutsumura.com/if-there-are-more-vectors-than-a-spanning-set-then-vectors-are-linearly-dependentMusic: Epidemic sound, Summer nights 2
Comment response video for Understanding Quantum Mechanics Looking Glass Universe 2018-04-24 | In this video I cover the common misconceptions I saw in my last video- Understanding Quantum Mechanics. In particular, I talked a lot about the measurement problem in this video.
Understanding Quantum Mechanics Looking Glass Universe 2018-03-22 | This video is an simple introduction to quantum mechanics that explains why there is still so much controversy over the meaning of this scientific theory. 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...
Announcing a new series of quantum videos Looking Glass Universe 2018-03-22 | A short update video.THE NEW SERIES HAS STARTED: youtu.be/8Dso6Fv1FUwHere 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/zDQH5x7svfgSuperposition ft. Schrodinger's Cat: youtu.be/zDQH5x7svfgWhat is a measurement?: youtu.be/KujEMGZGUTIQuantum Eraser: youtu.be/XcZ3jI1Ph7AQuantum Eraser explained: youtu.be/sQfSm6o-KlQDelayed Choice Quantum Eraser: youtu.be/MW-AemjSVGYHeisenberg Uncertainty Principle: youtu.be/ZpwZgOumTrsNew old series (from 3 years ago):Introduction to quantum mechanics: youtu.be/b_ddt6J1BioThe Wave Function: youtu.be/02eZMf17wFs(I will unlist some of the others later as I upload more in this new series.)
Do we have to accept Quantum weirdness? De Broglie Bohm Pilot Wave Theory explained Looking Glass Universe 2017-08-16 | An explanation and the pros and cons of Pilot Wave Theory aka Bohmian mechanics.Links to videos referenced:Veritasium's video: youtu.be/WIyTZDHuarQMy old Bohmian mechanics video: youtu.be/rbRVnC92sMsContextuality: youtu.be/Qz4CHI_W-TAEntanglement and the EPR paradox: youtu.be/Xzmp7byh77EAlso see: PBS spacetime's excellent video: youtu.be/RlXdsyctD50This amazing video about 'surreal paths' in Bohmian mechanics (this channel is also very worth checking out): youtu.be/CCW93koLNYY
Why Cant You Use Quantum Mechanics To Communicate Faster Than Light? Looking Glass Universe 2017-05-07 | 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.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.
How real are the real numbers, really? Looking Glass Universe 2017-03-03 | We usually say that infinity isn't real, but here we'll see how crucial it is to have one very big infinity for the real world; there is an infinite number of numbers. But why do we need real numbers at all? Aren't rational numbers enough? And what about hyperreal numbers?What we'll see in this video is that discovering or defining the real numbers is what allowed calculus to be made rigourous- and without it, we'd need to divide by 0 every time we took a derivative. This video is about the seeming mathematical paradox that arises to get Archilles from A to B (that isn't Zenos paradox!).Check out Nick Lucid's video on whether the universe is infinite: youtu.be/fApKpDGGDYk
Quantum Entanglement and the EPR Paradox Looking Glass Universe 2017-02-14 | What is entanglement really? And why is it that it's a uniquely quantum phenomena? Homework:1 Explain why Alice can never send any messages through this set up.2 This is one to practice working with wavefunctions of several objects. Prove mathematically that it’s impossible to write our state as two separate states for the electron and positron 3 This one is very relevant to quantum computing. If you have just one electron or positron, you only need 2 numbers to describe its state. Now suppose you have n of them. If they are not entangled, how many numbers to you need to describe the state? What about for a general state that is allowed to be entangled?Bohmian mechanics stuff:In Bohmian mechanics, you still have superpositions, they just mean something very different. A Bohmian mech particle only has one position, however, it still has a wavefunction that is a superposition of many possibilities. The difference is that the particle has one position but is influenced by other things in it's wavefunction. A particle like this behaves very differently to a classical particle that only has one position and is not at all influenced by it's other possibilities.
Resolution of the two envelope fallacy Looking Glass Universe 2017-01-29 | I talk about a very simple solution to the two envelope fallacy.Watch this video first: youtu.be/OqVFKY504X0
Solve The Two Envelopes Fallacy Looking Glass Universe 2017-01-23 | A fallacy arising from a surprisingly simple situation. Can you figure out what the problem is with this reasoning?
Why The Martingale Betting System Doesnt Work Looking Glass Universe 2017-01-15 | This is a follow up to a video where I described a betting system that seems to guarantee you win money- I asked you guys how that's possible. In this video I explain the flaws in the system. Previous video: youtu.be/t8L9GCophac
Can you solve this gambling paradox? Looking Glass Universe 2017-01-08 | Here's an interesting paradox for you to consider. We've all been told it doesn't pay to chase our losses gambling- but this video seems to prove that actually this will let you consistently win money. This can't be right- and that's the paradox. Solution announced at this time where you live: timeanddate.com/worldclock/fixedtime.html?msg=Gambling+Paradox+Answer+Video&iso=20170115T17&p1=136****************Hints:****************_______________Small hint:If you lost a lot of times in a row, you're going to run out of money at some point right? ______________________________Bigger hint:Say you have to stop playing if you ever lost x amount. What's your expected winnings?
