in practice, a lot of problems/equations can be approximated with so-called “linear equations”. a linear equation is something that can involve any number of variables, but no variable has any exponent other than 1. for example, 2x + 6y + 5z = 8 is a linear equation. (this is a generalization of the case when there are two variables, and you have an equation like 2x + 3y = 9. in this case, you can rearrange this equation into slope intercept form and get the standard equation of a line, hence the name linear.)

again, in practice, it’s very common to deal with multiple linear equations at the same time. say, 2x + 3y = 9, and x + y = 0. to solve such all equations simultaneously (in the 2 variable case) means finding x and y that satisfy both equations. and in the 2 variable case, it basically means finding a point where two lines intersect (if possible).

you can do some kind of advanced math to show that linear systems of equations correspond to matrices. this is “nice” because matrices are extremely easy for computers to deal with, and we also have a lot of theorems that talk about how matrices behave.

so, to summarize, we’ve reduced a real world problem into something involving matrices, with the hope of maybe having a computer solve it. in practice, many matrices can be “diagonalized”, which basically means you can factor it as a product of matrices satisfying some certain conditions, but i’m glossing over those details because it can be messy if you don’t know much linear algebra. you can think of it as kind of like factorizing a number into primes. it isn’t really the same thing, but it can be a helpful analogy maybe. (primes are easier to work with, and sometimes it’s helpful to view any number as just a bunch of primes multiplied together.)

the main advantage of diagonal matrices is that they’re very easy to work with (compared to matrices, which are already “nice” to work with). in practice, this is important because a lot of formulas, algorithms, etc only work for (or are most efficient on) diagonal matrices.

i hope this helps, it’s a bit hard to get into the details without making things needlessly complicated (a common problem for things involving matrices), but i tried to do my best to focus on the underlying concepts/real world use cases.

and why do you think they needed help with their math and reading homework? exactly…… because all they’re taught in school is how to be transgender

my phone charging port has been inoperable for years. wireless charging is the only thing keeping it alive

what are the “valid republican talking points” that you’re referring to?

i tried cast iron pans many summers ago. but i found myself never using it because cooking is already inconvenient enough and cast iron pans just add a whole other dimension of inconvenience. it also makes me nervous that they’re never/rarely ever cleaned with soap and water. it was also very difficult to find consistent instructions on how to care for them and use them safely. so now i just use stainless steel instead.

wood and stainless steel is definitely the way to go. i don’t understand how nonstick seems to be the norm. are people not aware that the “nonstick” part of their pans is basically just plastic? and that it’s generally a bad idea to cook/eat/scratch plastic things?

that would be bad, yes. but you could also make the argument that the guy in charge of things would be much less likely to initiate a war if he knew there was a chance he could get turned into beefaroni in a foxhole.

how does one consume ancient history? do you eat the source documents?

in conclusion, remember to always curse in your comments because it will make your code better

pretty incredible that they managed to get his signature even though he’s dead

how do you punch a wall into a toilet

sunk cost fallacy has got to be carrying trump a bit this election. it’s probably easier for some people to tune him out than to admit they’ve been wrong about him for the past however many years.

it seems to me like one of his greatest “strengths” was capturing the “i don’t care about politics” crowd. and for a decent chunk of those people, it’s maybe not so much about the politics as it is an alternative for something like professional wrestling or something. it’s about the spectacle and showmanship.

at least that’s what i keep telling myself. (i find it slightly more comforting than the alternative.) but don’t get me wrong, he definitely also has a ton of nazis supporting him.

can’t libgen get around this by just saying they’re training an AI model?

hopefully there’s a statue of limitations on this kind of thing or else you still might be in trouble

that doesn’t count though. the bible got grandfathered in

i’m not yet sold on this “old vs new” thing. while i do agree that it would be better if people were more engaged/active about boycotting things and pulling out the pitchforks, my understanding is that hasn’t been the historical precedent in situations like this. the pitchfork stuff certainly did happen quite a lot in the past, but my understanding is that it was for more extreme problems than a social media company shutting down third party apps (which many people didn’t even know about). but then again, it might be hard to compare this to the company transgressions of the past.

my understanding is that frustration is building, and if things continue in this direction, they will reach a tipping point where people do actually stop using the website all together and switch to alternatives. and, this ban on protests will give the reddit executives much less information on how close things are to that tipping point. (not to mention that the ban itself will probably accelerate things.)

but i could be mistaken about this, and i’m open to changing my mind on it.

we’ll be done with it exactly when the next fad picks up steam

with enough bombs they can blow up the next hurricane before it becomes a problem for the people living in florida

so many leaders are forgetting what the point of protests is. yes, protests are annoying if you’re a leader. but they’re better than the alternative. that’s the whole point.

i’ve mostly given up on expecting/trying to make coffee taste good. at this point i just eat the roasted beans and carry on with my day. it’s also more efficient since you lose a fair amount of caffeine through the brewing process. at this point i only have like 8-15 beans a day.

i still go to a cafe from time to time if i want a nice cappuccino or something, but i don’t bother with any of it at home.

edit: i should mention that roasted coffee beans can taste pretty good if they’re been roasted properly (and even better if eaten within a week of roasting). and you can get a pretty decent bean roaster for like 200$, and then after that a 3 lb. bag of unroasted beans costs like 30$. the 200$ upfront charge is pretty expensive, but it ends up paying for itself pretty quickly since you save an insane amount of money from buying unroasted coffee beans. when eating the roasted beans, a 3 lb. bag can last about 6 months to a year. so thats about 30$ to 60$ a year spent on coffee. even when i was brewing coffee, i found that the roaster paid for itself in about 6 months. not to mention that it’s not that hard to learn how to roast coffee beans, and everything tastes much better when using freshly roasted beans.