#multiplication of matrices

[kikaichuno]

//I just want to say, since I'm a biology nerd and my muse is too, I'm cool with having Shino answer and explain intricate biology/science concepts and will do it unprompted in replies too. BUT-

If I get a whole math problem in my inbox I will crash out- Shino will want to answer it but me? Have some mercy- I do math everyday- no way I'm gonna open my MATH or PHYSICS notes for a rp 🤣🤣🤣🤣 I will delete the ask.

[er-cryptid]

Matrix Multiplication - Ex. 2

Patreon

[xbuster]

Canon fact: Hibiki can solve for eigenvalues

What are they teaching this hamster?

So I was rewatching this scene and I happened to take a look at the book Hibiki is studying and...

Those are matrices.

Are they teaching linear algebra at Lydian? No wonder this hamster is always late on her assignments, that's like a junior-level college course!

[ghelgheli]

i would actually like to hear more of your thoughts on whipping girl, whenever you feel ready enough to talk about it. i've only ever heard positive recommendations for it. i was thinking of reading it. i've read one or two introductory 101 texts on transmisogyny as well as some medium/substack posts, and always looking to read more as a tme person. ty!

thanks for asking! I'm gonna try to be concise because I'm stuck on my phone for the month, but here are my thoughts on whipping girl:

serano is at her strongest in the book in three areas: manifestations of transmisogyny in media (e.g. how trans caricatures pervade movies), the history of medical institutions developing a pathology of transsexuality (like the diagnostics of blanchard et al. or how trans people seeking healthcare were and continue to be forced into acting out prescribed expressions and manufacturing memories), and the construction of her own transition narrative (telling the reader what it was like for her to grow up desiring femininity in a way that confused her, the experience of crossdressing, the effects of hrt for her)

whenever she's just sticking to this, I think she effectively communicates a lot that the unaware reader could benefit from—even many trans women/transfems/tma people who are otherwise in tune with the history of medicalized transsexualism and our popular depictions could probably benefit from her own personal narrative, by nature of how variegated our experiences can be.

unfortunately I think the book fails at its primary—stated—goal, which is to theorize about transmisogyny. in the big picture this is a bifurcated failure:

on one branch of her argument, she remains committed to there being something biologically essential/innate about gender. this manifests thru multiple claims: that we have "innate inclinations" toward masculinity/femininity and "subconscious sex" rather than what I believe, which is that the latter are constructed categories imposed on different matrices of behaviour/expression/desire in different cultural contexts; that there is "definitely a biological component to gender" (close paraphrase) after a discussion of how she believes E and T tend to affect people (thus equivocating gender with dominant hormones!); that we have such a thing as "physical sex" which is the composition of our culturally decided "sex characteristics" (don't ask me how the dividing line is drawn) even as she says we should stop using "biological sex" as a term; that there is "no harm" in agreeing that "sex" is largely bimodal with some exceptions; that social constructionism is necessarily erasure of transsexual experiences in early childhood... altogether she is unwilling to relinquish arguments about the partial "innateness" of femininity/masculinity and gender. this is at tension with her admission on several occasions that these are neither culturally/geographically nor temporally stable concepts! but that doesn't seem to be a line she can follow thru on.

on another, intertwining branch, she engages in what I think is a deep and widespread mistake in the theorizing of transmisogyny: reducing it (mechanistically) to what she calls effemimania* or essentially anti-femininity. it is her stated thesis at the start that masculinity is universally preferred to femininity. she doesn't offer a definition of either term until one of the final chapters, where she defines them as the behaviours and expressions associated with a particular gender. but I think this reduction just misunderstands transmisogyny. it is even in tension with an observation she makes early on, that trans women are often punished for their perceived masculinity! but again, this is a thought she seems unable or unwilling to follow thru with.

my problem with the thesis is that masculinity and femininity do not float free of gender—it is not possible to speak of their valuation in the abstract. anyone who grew up as a masculine cis girl and never "grew out" of that "phase" can attest to the violence wrought upon expressions of masculinity from women. and this applies doubly so to the subjects of transmisogyny! not only are we punished for any perceived bleed-through of masculinity from our supposed "underlying male selves", those of us who are willingly masculine and thriving as mascs are punished for our failure to conform to the rules of the normative womanhood that is imposed on us (just as we are punished for any willing femininity as "false" and predatory upon cis womanhood—observe that transmisogyny is reactive degendering in every case!).

on both branches serano makes only perfunctory remarks about the intersections with race, class, and colonialism. "sex" as such was made to only be accessible to the "civilized", most of all the white european! for a racialized person and particularly a Black person navigating gender the waters are just not the same; the signifiers of sex neither available in the same way, nor granted the same medical legitimacy. what is the "physical sex" of someone who is de-sexed altogether? how can gender have a "biologically innate" component when its expressions between the bourgeoisie and the working class are at total odds with one another? this all goes for the masculine/feminine distinctions as well. what sense is there in the claim that we have innately masculine/feminine inclinations when globally (and transmisogyny has been made global!) what is feminine and masculine can be very nearly mirrored? nor is "masculinity is always considered superior to femininity" innocent of obviating race. transmisogynoir adds yet further degendering thru the coercive masculinization of someone as a Black woman—masculinization as punishment, again!

and as a final point, the account fails to be materialist. there is no attempt to place transmisogyny in its role as an instrument of political economy or, as jules gill-peterson might say, as a tool of statecraft. it is just a psychological response to the way the world is, as far as serano has anything to say about it. but how did the world become that way, and why?? serano's solution, the abolition of what she calls gender entitlement, is naive to the fact that gender entitlement is necessary to the maintenance of the capitalist state, which is structured thru patriarchy and built on colonialism. it is not possible to reskin this into something innocuous!

this is why I cannot recommend whipping girl as a work about transmisogyny except at the most shallow level. it could be a helpful critical read, but imo, it is just wrong about transmisogyny.

[valdevia]

Nail matrix metaplasia is a cryptogenic disease where the nail bed starts growing, transforming the surrounding skin into multiple matrices that produce their own complete nails.

Though its origin is uncertain, it appears to be more common in people who bite their hangnails.

[lillified]

maybe a little bit too specific of a question. but. You mentioned that all cybertronians have "progenitors"- i was wondering if any of the original primes had contributed their parts to creating other bots? (follow up question if that's true: is it considered a "sacred bloodline" (energonline?)? are there small traces of the og primes in every cybertronian? something else?)

I love this question!!

Cybertronian lineages are a bit complicated—just to clarify the “progenitor” thing a little more, I’d say they’re sort of like fungus, coral, or jellyfish. When I say they contribute genetic material or parts to offspring, I mean that they literally do contribute physical parts of themselves which eventually grow into an entirely separate organism, and that reproduction can happen completely asexually, where the resulting Bot is a 1 to 1 clone, or “sexually,” where multiple ‘Bots contribute organic material to a single offspring resulting in not only greater variance, but a reduced negative physical impact on the “parents.”

While the exact historical timeline of most of the Primes is not known, it is concretely believed that, with one very notable exception, none of them have “ancestors.” Though they are known as originators, Cybertronians aren’t directly descended from them, and the early genetic groups were instead more like cousins or siblings to eachother. Those disparate groups eventually commingled and that resulted in the modern Cybertronian diaspora.

The reason for the Primes never “reproducing,” at least not publicly, is political/theological. There’s a lot of religious baggage tied up in the hierarchy of Cybertronian life where the Primes are the first and most perfect beings, so any one of them having a descendant would raise a lot of understandable questions about the Godhand and its implications on the rigid divine hierarchy’s mutability—in short, it would be a bit of a mess! The Matrices are a halfway solution to this problem, but before I get into those, I should probably mention the Notable Exception.

So, there is one Prime with technical “offspring,” and that is Quintus. The Quintessons are Quintus Prime’s creation, and though they were deliberately engineered in a process that is quite a bit more complicated, they do have his CNA. Their existence very famously led to the end of the feudal hierarchy and a new era of caste-based determinism where social value and divine right were assigned via function. The Quintessons brought about an era where old flesh was no longer immutable and absolute, opening the floodgates for the philosophy of mechanization and new flesh. This (and the fact that, well, most of the original Primes are dead now) is why he is the exception—every acting Prime knows that the Godhand is dangerous.

That said, though, there is certainly an impetus for successors, and that’s where the Matrix came about. Because the exchange of genetic material isn’t always 1 to 1, you don’t have to be a direct parent to pass on material to another ‘Bot, or vice versa. In the right conditions, with the right material, a Cybertronian can absorb somebody else’s material to become genetically similar to them.

A Matrix is an isolated piece of an original Prime. While it’s hardly enough to overwrite the host organism, the Matrix bearer essentially integrates that material, absorbing the polyp sort of like a parasitic twin in utero, and technically changing the bearer’s CNA. So, in that sense, you can say all Primes are descendants of the originals!

Sorry for the long response, I hope this gave you the info you were looking for! I loved this question and I’ve put a lot of thought into this subject :)

[acekoral]

That's also what I love numpy for, it's a great feature that has no drawbacks at all and can never lead to confusions

[dreamylove444]

🌸Study log after school 26/11/2024 (3h20)🌸

⊹ ⁺ 𐔌 ᩧ ຼ ͡ ৯ ♡໒⁀ ᩧຼ ꒱ིྀ ⁺ ⊹

🌷 revision physics

🌷2 Kepler law exercices

🌷revision chemistry

🌷2 acid base reaction exercises

🌷revision matrices multiplication substraction addition

🌷2 matrice exercices

⊹ ⁺ 𐔌 ᩧ ຼ ͡ ৯ ♡໒⁀ ᩧຼ ꒱ིྀ ⁺ ⊹

⊹ ⁺ 𐔌 ᩧ ຼ ͡ ৯ ♡໒⁀ ᩧຼ ꒱ིྀ ⁺ ⊹

[la-cosmonauta-extraterrestre]

How to learn physics as an adult

I'm creating this post in response to some posts @un-ionizetheradlab made the other day, but I'm creating this as a guide to anyone this is relevant for. It's going to be a long post, but pick and choose what to do from this list based on what works for you and what your goals are, whether it's just to gain basic scientific literacy or become a physicist (or something in between). Also remember that it's a journey not a sprint, so it's ok if you don't understand physics at first (and if it makes you feel better, physics was one of my worst subjects in school and now I have a master's degree in physics). Without further ado:

First Thing's First

There are some mathematical methods you need to learn to understand physics; there's no way around this:

Vectors: This is the most important thing to learn for physics, how to use vectors. It seemed every mathematics or physics class I took in my first year of my physics degree started with an introduction to vectors, and for good reason. You can learn about how vectors work on Khan Academy for free.

Matrices and Tensors: Once you've mastered vectors, learn about matrices and linear algebra, and perhaps go on to learning about tensors once you're at it. You can at least get the basics about matrices from Khan Academy, but you might want to invest in a linear algebra textbook.

Calculus: I said vectors are the most important thing to learn for physics, but it actually might be calculus. If you have absolutely no previous knowledge of calculus, you can watch the video "Calculus at a Fifth Grade Level" on YouTube; it's a little more advanced than fifth grade level but can give you a good feel for what calculus is about. Once you've done that, there are multiple calculus courses available on Khan Academy. There's also a calculus course available on Brilliant, but it might only be available through the paid version.

Ordinary Differential Equations and Boundary Value Problems: You don't need to learn these right away, but if you want to do physics at the upper undergraduate level, you'll need to learn these at some point.

2. Learning to Think Like a Scientist

Some suggestions of apps and things to watch if you don't know much about science so you can start thinking like a scientist:

SciShow: If you don't know much science at all, SciShow on YouTube is a good place to start, I used to watch it and as I recall it's more focused on life sciences but there's some physics videos there, too.

Ciencias De La Ciencia: This is sort of a Spanish version of SciShow but it's more physics-focused. At least some of the videos have subtitles if you don't know Spanish.

Cosmos: If you haven't seen Cosmos (either the old version with Carl Sagan or the new version with Neil DeGrasse Tyson), it's very good and at least some of the episodes are available for free online. It's more pop-science and history of science than actual science content, but at least they make a point of using anecdotes from the history of science to illustrate how the scientific method works.

Sabine Hossenfelder: Highly recommend her YouTube channel; she's one of the most intellectually honest scientific communicators in the world nowadays. Her videos are a good illustration of how to think like a scientist. She also has a blog and has written a few books.

Brilliant: This is an app with mathematics and science courses that places an emphasis on problem-solving. Most of the courses are only available on the paid version of the app (but you should be able to get a discount on it if you're subscribed to any mainstream science YouTubers), but even the free version gives you access to a few courses, plus a forum where people post problems. I had this app back in the day and liked solving problems on the forum (no idea if it's changed since then).

3. Books to Study

If you're committed to learning physics you should study from some textbooks:

Physics LibreTexts: This is a whole collection of university-level physics textbooks for free online. It's an invaluable resource for learning physics. Use it to learn classical mechanics, electricity and magnetism, modern physics (but don't jump into quantum mechanics straight away if you're just starting out in physics).

Landau and Lifshitz Course of Theoretical Physics: This was the physics school curriculum in the Soviet Union; it's a little dated now but if you're just learning the basics it can't be beat given the excellent pedagogy. It's easy enough to find copies of it online, especially on Russian sites. Most if not all of the textbooks in the series have been translated into English, but if you know any Russian, the original is easy to follow.

Introduction to Electrodynamics by David J. Griffiths: Once you've learned mechanics, modern physics and some electricity and magnetism get yourself a copy of this textbook; you can get used editions on Amazon for a reasonable enough price. American physics majors are obsessed with this textbook, refer to it as the Bible, and for good reason.

Every Life is on Fire by Jeremy England: This isn't a textbook, but reading it took me back to my statistical mechanics class and it's way more readable than any actual statmech textbook so if you are interested in learning statmech, this book is a good start. It's actually a general reading book about England's ideas about the origin of life, interspersed with some parallels to the Hebrew Bible because England is also a rabbi. He actually has some interesting ideas about the philosophy of science, though they can be difficult to get behind, so if you're interested go listen to a podcast where they interview him (obligatory I don't condone the Kahaneist politics he sometimes promotes).

4. Learn About Research and Experiments

Physics is an experimental science, so expose yourself to some experiments:

Look for PDFs of high school physics labs online. You can find some for free and it should be cheap enough to do the experiments at home.

Read scientific papers on topics that interest you to try and understand what's happening today. If you find them difficult to understand, try reading older papers and go from there, for example, in undergrad I did a research internship relating to neutron stars, but I found some of the recent scientific papers difficult to understand, but reading the 1938 paper "On Massive Neutron Cores" by Oppenheimer and Volkoff helped me to understand neutron stars better. (When I returned to some of those same papers during my master's degree, I was proud to have understood them well.)

5. Take University-Level Physics Courses

You can take university-level physics courses without committing to a degree:

Search online for MOOCs (Massive Open Online Courses). You can find MOOCs on multiple sites about many physics topics, and they're often free (sometimes you have to pay for them).

If you live in the United States, you can take physics classes at your local community college.

You can enroll in online physics courses through Open University, based in London but you can take the courses from anywhere. It's expensive, but you pay by the credit so you don't have to pay for a whole year of tuition if you're just taking one course.

If you happen to have free time in the summer and the money for it, many American universities (and elite British universities) offer summer courses that one can enroll in even if they don't attend the university. These are usually in-person classes.

6. Get a Physics Degree

Getting a physics degree is ultimately the only course of action if you've decided to become a physicist, the recommended course of action if you're ultimate goal is a PhD in the history of science or philosophy of science, and a good idea if learning physics has made you want a career in science communication of science education. There's no shame in being a non-traditional age student; in both my bachelor's and master's degree in physics I knew students who were non-traditional age. The downside of this is that it's a bad financial decision to get a degree, especially if it's a second bachelor's degree, but there are ways to lessen the financial burden of a degree:

If you attend an American university with American tuition, you can usually get an on-campus job, though that's pocket change compared to the costs of tuition.

On the bright side, if you already have a bachelor's degree you can probably get credit for general requirements at American universities, so a second bachelor's degree in physics might not take long.

You can also do a part-time degree while you work at many universities.

Just some general advice, if you go the American university route go to a university with a Society of Physics Students and get a student membership in American Physical Society; you get all kinds of benefits like access to Physics Today magazine, scholarships, internships, conferences, an honour society induction.

All that said, it's difficult to attend an American university without losing money. For that matter it's difficult to attend any university in the world without losing money, but you can lessen that burden by going to a country where university is cheaper. There the limiting factor is going to be language; although English is the international language of physics and the medium of most postgraduate physics degrees around the world, physics bachelor's degrees are usually in the local language. Some possible exceptions I found to this, for those who are not fluent in a language other than English:

Apparently there are world-class English-medium physics degree programmes in France? I figure there must be some kind of catch given the way the French are about their language, but given the high research output France has in physics, this is worth getting into.

There are English-medium physics bachelor's degrees in the Czech Republic, and tuition there is pretty affordable (for the English-language degrees; it's free for Czech-language degrees if you happen to be fluent in Czech). I don't know Czechia to have a lot of physics research output today, but back in the day Prague was a major centre of scientific research (Einstein briefly lived and worked as a physics professor in Prague), so it you're goal is to do a PhD in the history of science...I'm just gonna say that there's an English-medium physics bachelor's degree programme at Charles University and you'd have time when you're not studying to explore the city and it's history (but you should learn some Czech if you're going to live there).

University degrees in South Africa are usually English-medium, and tuition there is pretty affordable. There's also a fair amount of research output from South African universities. (Though I understand not wanting to live in South Africa.)

[allsoundwavesarebeautiful]

i lightly skimmed the MTMTE notebooks and there's a lot of interesting stuff in here. for instance: Cyclonus, Whirl, and Tailgate were at some point intended to be explicitly canonized as a polyamorous couple (the end result is more ambiguous, and i think Roberts said on some other occasion that he thinks Whirl became Cyclonus's amica after the finale [i think being the operative phrase here don't quote me on this], so evidently he wasn't super committed to this idea)

Drift was originally a proto-Getaway who was going to manipulate Rodimus out of his captaincy & take over the ship bc he was obsessed with finding the Knights of Cybertron and Rodimus's diversions aggravated him

Ultra Magnus was going to leave the ship on multiple occasions bc his motivation was that he was hunting down rogue Decepticons, and he'd come back every once in a while. i don't recall any of that making it into MTMTE

Functionist Universe Rung was supposed to meet up with Original Rung during the penultimate battle against the Functionists and they were going to die together making 6 Matrices each. i'm particularly glad this one didn't materialize bc i think that would have just been too much, man. i was sad enough when 1 Rung did it

and Rung was planned to be Primus from very early on. Rung was the reason i started skimming the notebooks in the first place bc i wanted to see if i cld glean anything abt author's intent wrt the idea of divinity; there wasn't a whole lot but i don't think my whole thing abt gods being socially constructed was an intended takeaway? which doesn't change my mind at all

i didn't read through very much of them so that's all i have that interested me. i thought it was strange that there were zero character notes on Cyclonus in the Character Platforms sections though

[maths-terms-identifier]

Module

Abstract Algebra

Let R be a ring, then a left R-module, M is an Abelian group (M,+) along with an operation ·:R×M->M such that for all r,s∈R and x,y∈M we have

r·(x+y)=r·x+r·y,

(r+s)·x=r·x+s·x

(r·s)·x=r·(s·x)

1·x=x

+ is called addition and · is called scalar multiplication

trying to learn tech is so fucking hard there's so much jargon. "what gooble are you using" "what's a gooble" "you use the gooble to respink your matrix" "what does respink mean and what is a matrix" "respinking means unpiling a component to change it back to its base spink. a matrix connects your device's AKD to your networks z jamper" how does anyone understand any of this

[lunarsilkscreen]

4D is 3D, Actually.

This one I'm not gonna be able to prove on my own; just a simple observation while uh... Perusing 3b1b videos.

The ones relating to the concepts they strung together in video *actually*.

3b1b did a video about [math puzzles for thinking outside the bun] and inspired this whole thought process.

Take a look, after watching the whole video, at the section about Quaternion Rotation. Look at how the rotation of the sphere makes the 4D sphere (which is a donut actually)

It's of my, intuition, that this is simply rotating the perspective of the object.

So instead of the object rotating around a focal point; the focal point itself is rotating *through* the object.

This accompanies my observations on 3D-rotational matrices (and vectors) surrounding 3D objects. (x,y,z) Where x and y determine the rotational planes, and the z value is a combination of both (like a diagonal... I lack the correct terminology but effectively;

{z...} = {x...} × {y...}

This suggests that when working in 4D space; as opposed to 2 and 3 which are clearly defined axis; the 4 dimensional axis (in real time) follows a set position, *and then* rotates itself; with each subsequent value over time wholey dependent on the trans-dimensional rotation.

Effectively; dimensional maths starts breaking down at higher-maths; because the pointer-vector(focal point){whatever you wanna call it} shifts back inward on itself.

Like how each {x,y} in y=mx+b draws a line, but only after multiple (different) x/y values have been passed into the function.

[abalidoth]

what was your PhD dissertation about?

No idea what your level of math is so I'll answer this at a few levels.

Level 1:

Group theory is the study of symmetries of objects -- like how a rectangle is symmetric in the same way the letter H is symmetric, but not in the way the letter S is symmetric. I studied the symmetries of very specific sets of points that live in many-thousand-dimensional space.

Level 2:

I studied a type of object called an association scheme that can be seen either as related to a finite group, or a special type of graph, or a special matrix algebra. My specific research involved digging through a database of finite groups to find schemes with a particular useful property, and proving some generalizations of it.

Level 3:

An association scheme is a special kind of matrix algebra, that's closed not only under regular matrix multiplication but also elementwise matrix multiplication, and has a basis of 0-1 matrices. These basis elements form a series of graphs with special properties -- strongly regular graphs and distance regular graphs are both a type of association scheme. Association schemes can also be developed from a generously transitive group action (one where any pair of elements has a group element that switches them.)

My research involved proving properties from the character theory of finite groups that allows me to probe into the structures of association schemes, and dig through old databases of finite simple groups to prove shit. There's a number on my forearm tattoo (13056) that's the dimensionality of a new association scheme I discovered lurking inside the Co2 (Conway) finite group.

[bubbloquacious]

@locally-normal asked me: 16. Can you share a good math problem you’ve solved recently? From Real's Math Ask Meme.

So for a group project I have been looking into group schemes, which are a generalization of algebraic groups, i.e. groups whose underlying set is locally like the set of solutions of an algebraic equation. One way of taking the abstract space that is a scheme to something concrete is its functor of points, which takes in a commutative ring A, and spits out the set of 'A-valued points', defined as the set of scheme morphisms Spec A -> X, denoted X(A). For example, if X is the affine plane scheme (explicitly, X = Spec ℤ[x₁,x₂]), then the X(A) is in a natural bijection with the points of A².

A group scheme is a group object G in the category of schemes, which means there are scheme homomorphisms m: G ⨯ G -> G, e: Spec ℤ -> G, s: G -> G which behave like the multiplication, unit, and inversion operations of an ordinary group. In particular these morphisms functorially induce a group structure on the set G(A) for every ring A. But it goes the other way too! If you have a functor from rings to groups whose underlying functor to sets is the functor of points of some scheme G, then this induces a unique group scheme structure on G.

The category of affine schemes, which are exactly those schemes isomorphic to the spectrum of a ring, is dual to the category of rings. This means that the scheme morphisms Spec A -> Spec B correspond naturally to ring homomorphisms B -> A and vice versa. So if Spec A comes equipped with a group scheme structure, then the scheme morphisms m, e, s when transported to ring world turn into ring homomorphisms μ: A -> A ⊗ A, ε: A -> ℤ, S: A -> A. The equations that they satisfy are exactly the same equations that make μ, ε, and S into a (ℤ-)Hopf algebra structure on A.

So if you have a commutative ring A, then the general linear group of degree n is the group GLₙ(A) of invertible n ⨯ n matrices with entries in A. A ring homomorphism A -> B induces a group homomorphism GLₙ(A) -> GLₙ(B), which makes GLₙ into a functor from rings to groups. This functor is (naturally isomorphic to) the functor of points of the spectrum of 𝒪(GLₙ) = ℤ[x₁₁,x₁₂,...,xₙₙ,1/det], i.e. the polynomial ring in n² indeterminates localized by inverting the polynomial that gives the determinant of the matrix (xᵢⱼ)ᵢⱼ. To see this, note that a ring homomorphism 𝒪(GLₙ) -> A (i.e. a scheme morphism Spec A -> Spec 𝒪(GLₙ)) is uniquely determined by the images aᵢⱼ of the generators xᵢⱼ, and the only requirement of these aᵢⱼ is that the determinant of (aᵢⱼ)ᵢⱼ is a unit. In other words, this matrix is an element of GLₙ(A). What are the Hopf algebra homomorphisms on 𝒪(GLₙ) associated to its group scheme structure? Let's work it out for degree 2.

This is the construction: to get the comultiplication μ, you take the group elements associated to the tensor product injections (the tensor product is the ring world equivalent of the product in scheme world), which for GL₂ are [[x₁₁ ⊗ 1, x₁₂ ⊗ 1], [x₂₁ ⊗ 1, x₂₂ ⊗ 1]] and [[1 ⊗ x₁₁, 1 ⊗ x₁₂], [1 ⊗ x₂₁, 1 ⊗ x₂₂]]. Then you take the product of these group elements, and convert the result back into a ring homomorphism. This gets us

μ(x₁₁) = x₁₁ ⊗ x₁₁ + x₁₂ ⊗ x₂₁, μ(x₁₂) = x₁₁ ⊗ x₁₂ + x₁₂ ⊗ x₂₂, μ(x₂₁) = x₂₁ ⊗ x₁₁ + x₂₂ ⊗ x₂₁, μ(x₂₂) = x₂₁ ⊗ x₁₂ + x₂₂ ⊗ x₂₂.

To get the counit ε, you take the unit element of the group, which for GL₂ is [[1,0],[0,1]], and turn it back into a ring homomorphism. So

ε(x₁₁) = 1, ε(x₁₂) = 0, ε(x₂₁) = 0, ε(x₂₂) = 1.

Finally, for the antipode S, you take the group element associated to the identity homomorphism, which for GL₂ is [[x₁₁,x₁₂],[x₂₁,x₂₂]], invert it, and then turn it back into a ring homomorphism. So if we remember the formula for inverting a 2 ⨯ 2 matrix, we get

S(x₁₁) = x₂₂/det, S(x₁₂) = -x₁₂/det, S(x₂₁) = -x₂₁/det, S(x₂₂) = x₁₁/det.

So I thought that was pretty interesting :)

[onlinesolve]

"Need to calculate matrices quickly and easily? Check out our online matrix calculator! Supports multiple languages. Click the link below! https://onlinesolve.net/en/matrix-calculator/

#Matrix #mathematicsmadeeasy #matriks #mathematics #math #matrix #matrices #matrixcalculations #onlinecalculator

[shankhachil]

Countdown to JEE (Main): Week 4/33

This week's been a bit stressful because of unit tests and also because I haven't been very well, what with the weather change :') I did my best though!!

Test results:

Allen monthly test: 205/360, rank 2/8. In my defence, the questions were really hard and there were a few chapters I hadn't studied yet....

Topics covered:

Physics: Electrostatics; Electric Potential and Capacitance; Simple Harmonic Motion; Fluids; Modern Physics (5/3)

Chemistry: Alcohols, Phenols and Ethers; Halogen Derivatives; Isomerism; Ionic Equilibrium (4/3)

Mathematics: Matrices; Vectors; Differential Equations; Permutation and Combination (4/3)

Questions solved:

Physics: - Physics tuition question bank, Electrostatics and Electric Potential and Capacitance — 123 questions, 114 correct - Allen Electrostatics module, S — 28 questions, 25 correct - Physics tuition Potential and Capacitance module — 38 questions, 35 correct - Physics tuition Electrostatics module — 31 questions, 27 correct - Allen Student Question Bank, Simple Harmonic Motion — 40 questions, 32 correct - Allen Student Question Bank, Fluids — 20 questions, 20 correct Total: 280/60 questions, 253 correct

Chemistry: - Allen Organic RACE 1,2,3,4,5 — 50 questions, 43 correct - Allen Physical RACE 11, 12, 13, 14 — 40 questions, 34 correct - FIITJEE Alcohols, Phenols and Ethers module, Exercises 1, 2, 3, 4, 5 and 6 — 48 questions, 42 correct - Kota Question Bank, Halogen Derivatives, single-choice, multiple-choice, comprehension and matching-type — 162 questions, 138 correct - Kota Question Bank, Alcohols, Ethers and Epoxy, single-choice — 70 questions, 55 correct - Kota Question Bank, Isomerism, single-choice questions — 51 questions, 42 correct Total: 421/60 questions, 354 correct

Mathematics: - Cengage Algebra Matrices, multiple-correct questions — 33 questions, 26 correct - Allen Differential Equations module, O3 and O4 — 20 questions, 16 correct - FIITJEE JEE (Advanced) archives, Permutation and Combination — 31 questions, 26 correct - Yellow Book, Permutation and Combination, single-choice — 26 questions, 24 correct Total: 110/60 questions, 92 correct

GRAND TOTAL: 811/400 questions, 699 correct

I!!!! Still!!!! Need to work on my accuracy!!! it's so frustrating but I promise I am Trying My Best

Upcoming tests:

28/06 (Friday) — Test at physics tuition center (JEE (Advanced) pattern). Topics: Motion in One and Two Dimensions; Units and Dimensions; Electrostatics; Potential and Capacitance; General Organic Chemistry; Halogen Derivatives; Chemical Thermodynamics and Thermochemistry; Alcohols, Phenols, and Ethers.

30/06 (Sunday) — Online test for mathematics tuition (JEE (Advanced) pattern). Topics: Trigonometry; Logarithms; Quadratic Equations; Sequences and Series; Trigonometric Equations; Permutation and Combination; Functions; Limits, Continuity and Differentiability

See you again next week!