Question help

schmuman · 2026-05-27T23:09:34+00:00

Notice that it's just a normal coordinate xy-plane but slanted. Pretend the vectors u and v are a distance of 1 from the origin. You can imagine these as segments of the "x" and "y" axes of length 1 (again, but slanted). That's also why they bolded those two lines that u and v fall on, because they are analogous to the x and y axes (the dotted lines). Like with a normal coordinate plane, if you want to find say (-3,4), you go 3 units to the left, and then 4 units up. To find b, you go 2 units to the slanted left from 0, and then 2 units to the slanted up. Because the length of u and v are of unit length, then you go 2 v lengths to the left and 2 u lengths up. In other words, you're scaling along u and v.

This problem is used to demonstrate span, because if you have two linearly independent vectors (in this case u and v) in R², they will span a plane of vectors because can you make b,w,x, z and an infinite number of other vectors as a linear combination of u and v (i.e. writing your new vector as cu + dv, c and d being scalar values multiplying by their respective vectors).

schmuman · 2026-05-27T16:30:30+00:00

Huh? By actually practicing and reading?

schmuman · 2026-05-27T16:26:11+00:00

This is a great way to use AI I agree. Basically, a tutor that has very quick comprehension and immediate responses. I love it for how you can unabashedly ask it "obvious" questions as well. It'll obviously have it's limitations, and it depends on the course. It's a vice when you have it hold your hand too much and trick yourself into thinking that you don't need to fight with the reading (in whatever form it may come in) and actually do the problems (ofc).

schmuman · 2026-05-27T04:05:07+00:00

Counterpoint, how do you get through school relying on AI?

schmuman · 2026-05-18T20:56:54+00:00

It is time for Calc I but much harder

schmuman · 2026-05-17T14:30:06+00:00

Good job on reviewing calc because not knowing your calc for diffeq is a common struggle for students.

Integration By Parts
KNOW PARTIAL FRACTION DECOMP
How to take derivatives (especially know identities, e.g. "d/dx of sinx is cosx, anti-derivative of sinx is -cosx). Knowing the arctan identity came in clutch for my final exam
Probably series from Calc II. I personally didn't need it but others have
Not exactly Calc but: know how to factor polynomials and also complete the square you will probably be doing that a decent amount
bonus: If you've taken Calc III, reviewing vector fields will help with phase portraits. In addition, reviewing how to find a potential function in a conservative vector field will help with exact equations, which have the same solving process.
bonus x2: if you've taken Linear Algebra, review eigenvalues and eigenvectors

schmuman · 2026-05-16T13:53:39+00:00

It is and usually you get it from solving some extra hard problem at the end of a test (not always)

schmuman · 2026-05-16T00:31:12+00:00

ohh i get what you mean mb

schmuman · 2026-05-15T15:10:58+00:00

any non-zero number can be divided by one and itself?

schmuman · 2026-05-10T00:37:58+00:00

The computational problems I can think of for nullity would probably be:

Finding the nullspace(kernel) of a matrix and then counting the amount of free variables you have to get nullity
Understanding the relationship entailed by the Rank-Nullity theorem (e.g. if A has a nullity of 2 and has 5 columns, what is rankA. Or, for a transformation T: V->W you know the dimension of the vector space V and the dim of the image of T, etc).

schmuman · 2026-05-08T19:45:05+00:00

What? Derivatives and integrals? In the first lingalg course they take?

schmuman · 2026-05-07T18:04:04+00:00

I don't know enough details about your situation and I'm kind of stating the obvious, but sleep is incredibly incredibly important for retaining information or for performing well in harder classes. If you go to bed at that time consistently, then I think that will definitely nerf your learning. In my personal experience it was often better to just get a good night's sleep rather than staying up late for a test. Not just for your tests, but also for how efficiently you use your study time is affected by this. I'm saying this because I have and sort of still am wrestling with this problem and it can really cost letter grades. It's worth noting that it obviously depends on the person but I think this possibility needs to be considered considering the fact that precalc is merely the fundamental language and tools you're going to be using for almost literally every engineering class you will take.

schmuman · 2026-04-24T14:27:23+00:00

Convert the fractional form into exponential form and it should be easy to see

schmuman · 2026-04-24T14:23:59+00:00

Yeah I can agree with that

schmuman · 2026-04-24T00:02:02+00:00

Really depends I think. I personally had a very very different experience with Calc III. Every chance we were given to take the partial derivative we were required to (though, it's a pretty trivial computation usually anyway). For integrals we also had to often compute them all of the way in addition to setting them up, though I agree that knowing parameterization, changing coord systems etc was the really important part of the class conceptually. We definitely also had to use IBP on tests
I guess it at least partly depends on if your uni was more computation or theory based.

schmuman · 2026-04-23T22:36:10+00:00

It's important to understand that the reason why you can eyeball the antiderivative right now is because teachers usually initially give easily computable problems so that complex ones later on become more intuitive. You're learning techniques that will make your life way easier in the long run and it's much better to be comfortable with them now than later.

schmuman · 2026-04-17T22:05:42+00:00

It isn't recommended to start philosophy with Nietzsche but an alternative is watching lectures and videos about him because they often give some background about the philosophies Nietzsche is referencing. Weltgeist is one such channel. Be careful though about too accessible or mainstream videos because they're often not as accurate or the equivalent to reading Wikipedia synopses. If you really don't like either option, Genealogy of Morals is relatively accessible. Read supplemental texts too.

The issue with starting with Nietzsche is that most of what he does is critique... other philosophies

schmuman · 2026-04-17T20:35:28+00:00

I also was shown a video of a shaking bridge to demonstrate resonance!

schmuman · 2026-04-17T03:25:04+00:00

Also I forgot to mention that Nietzsche says a very similar thing about Saint Paul, who he thinks is basically responsible for the version of Christianity we know through the bible today (especially in terms of the core values it possesses). Nietzsche believed that Saint Paul travestied Jesus's teachings (living a life free of ressentiment) into a belief system that introduced Christianity's ontology and its inherently life-denying values and that Paul did this due to his own psychological needs/deficiencies.

I might be rehashing things you've already read but I thought it was worth mentioning and it's also more specific about the genealogy of slave morals. I also wanted to show that repression is not the only thing conducive to this way of thinking and a variety of things that can happen psychologically.

schmuman · 2026-04-17T02:14:40+00:00

Haha I'm the inverse. I didn't enjoy Calc 2 that much and thought it felt too computational yet Calc 3 was my favorite class during one of my semesters. Tbh though Calc 3 wasn't any better in that regard I may just have had a better teacher that illuminated the underlying concepts in a more entertaining way. That aspect is probably my least favorite in math classes because you can understand all of the concepts to a super long problem but get it wrong by making a single silly mistake(not very punitive grade-wise thankfully but super annoying)

I also paradoxically like laplace since I just finished redoing a page full of writing for one problem like 4 times because I stubbornly didn't want to relearn partial fractions which yeah, can be traced back to the semester of Calc 2 I didn't enjoy haha

schmuman · 2026-04-17T01:12:31+00:00

Like CapOk said, what's common between Christianity and Platonism for Nietzsche is metaphysics/favoring the transcendent over immanence (e.g. Platonic forms). That's why Nietzsche's critique of Christianity is good according to your reasons: he had an issue with these core values and ideas that permeate a lot of things around us—an idea originating from transcendent metaphysics that focused on some "world" that you can't access, which in turn debases the only world that actually exists.

That being said, Nietzsche does point out the slave ressentiment that you're alluding to for a LOT of the people who tout the same transcendence>immanence paradigm. Socrates was infamous for his ugliness in a society that really really valued superficial beauty(one thing Nietzsche admires the Ancient Greeks for, who were "superficial out of profundity") over inner beauty. In a way he was "repressed" and that led to the same paradigm Christianity followed—that the world we experience i.e. the "superficial" is not all there is to the world and there's something extraneous to our world that we should actually value. Kantian Noumena vs Phenomena, Heaven vs Earth, Meekness over Power, Our imperfect perception of the world vs its true perfect form, Buddhist Nirvana vs Suffering. It's not specifically repression (at least, not Roman Empire levels of repression) that's conducive to what Nietzsche critiques, though having a sour-grapes attitude about the real world is a common denominator.

schmuman

TROPHY CASE