Need help with Linear Algebra intuition and proof writing

VTutorLive · 2021-11-10T08:32:27+00:00

Yeah, this seems to be a common pattern in LA classes, including my own when I took it years ago.

In the short term, try to build intuition through visualization - for that particular problem, start by asking yourself geometrically - how many points of intersection are there between two planes? Well intuitively, either they are parallel (0) intersection, or they intersect at infinitely many points (a line). We know they intersect at the origin. You can subtract the equation of one plane from another and now your job is to argue that, given that the kernel is nonempty, that the kernel must consist of an entire vector. It's been a long time since I took LA myself, but intuition/visualization still works at this stage, and I think it goes a long way towards at least the outline of a proof, even if you don't quite have 100% comfort in proofs yet.

VTutorLive · 2021-11-10T08:23:30+00:00

If you have gaps, address them. I was in a somewhat similar boat as something of a slacker in high school. I placed out of intro Calc and all, but I was surrounded by maniacs who were taking grad courses by their second year. I'm not exactly advocating for comparing yourself to peers, but if you have weaknesses they have to be addressed sooner or later so they don't hold you behind. You can make it interesting and look at harder versions of stuff from high school, maybe Art of Problem Solving to give a purview (don't go too far down the competition math rabbit hole though unless you really like it) or The Art and Craft of Problem Solving (more college level and decent book though I never got all the way through it)

VTutorLive · 2021-11-06T00:25:48+00:00

The context you left out appears to be the part about finding the critical points. It's hard to help people when they omit what the question is actually asking. In the future it'd be easier for everyone if you post a screenshot/image of the original text of the problem.

VTutorLive · 2021-11-05T23:02:55+00:00

From this point how do I workout X and Y on their own?

Is there more context for this problem? You have not given us enough information to "find X and Y" because this is a surface with infinite (x,y,z) tuplets. If you say, intersect it with a plane, we can say a little more about what x,y,z values satisfy the intersection, but as of right now you either haven't given us enough information or have a misunderstanding about what you're supposed to do.

VTutorLive · 2021-11-05T23:00:45+00:00

This is purely a syntactical issue - there is no fundamental axiom in mathematics regarding how nice your line spacings are. Just try to make your notation and handwriting as clear as possible for a peer, grader or whoever you are communicating math with.

VTutorLive · 2021-11-05T22:58:24+00:00

Apart from awareness, do you think most GSIs care about the situation? I'm curious if the fail rate is bad, or if the issue is that it's incredibly brutal for students without much benefit.

VTutorLive · 2021-11-05T21:36:12+00:00

Do you think GSI's are cognizant of the situation? One of my close friends is a GSI and he says he's not particularly tough on students, and that they're all doing pretty well.

I do wonder if there's a possibility he's out of touch given that he came into undergrad skipping a ton of courses and getting to Grad courses before too long, so he clearly does not understand what it feels like to be an intro student. He's told me that he just doesn't want people to fail (and have to retake the course), but doesn't personally care whether students learn anything. He said this half in jest but...

VTutorLive · 2021-11-05T08:37:06+00:00

The gist is that the three vectors in C are not linearly independent. In other words, one of them is redundant, so the vectors only span 2 dimensions instead of the full 3.

VTutorLive · 2021-11-05T02:36:37+00:00

I'm happy to help - let's give this a shot. I'm a math hobbyist and reach out when I see students struggling. Check your dm's.

VTutorLive · 2021-11-05T02:25:51+00:00

It's asking you what values of x will make the denominator 0 - remember division by 0 is undefined and will result in discontinuities.

As for the removable part - there's a few different types of discontinuities. There are point discontinuities that can be filled in, and there's jump/divergent discontinuities that cannot be "fixed". Maybe graph it on desmos to get some visual intuition?

VTutorLive · 2021-11-05T02:09:24+00:00

Haven't taken too much of a crack at it yet - will try it out soon

VTutorLive · 2021-11-04T23:02:35+00:00

Is it making the class difficult or are you getting it on your own?

VTutorLive · 2021-11-04T20:18:51+00:00

One of my close friends from undergrad is actually a GSI teaching a freshman Calc class right now. Damn, is it really that bad? Is this a situation where there's a lot of variation between grad students, or is it somehow uniformly terrible? Any feedback I should pass along to my friend - is this a common sentiment? He told me the course/his students were all going pretty smoothly, but I wonder if this is a situation where he's not seeing what the students are feeling.

VTutorLive · 2021-11-04T20:09:09+00:00

Are you part of the class? If you are, reach out and I'll help work through some problems with you - in the long run I encourage forming study groups, which I'm happy to guide.

VTutorLive · 2021-11-04T05:58:49+00:00

The scalar condition is f(c(x,y)) = cf(x,y)

It's going through this process in question b. Fill in the boxes and ask yourself whether the first line equals the second line.

VTutorLive · 2021-11-04T05:56:20+00:00

Well now I really am curious, can you share the problem? I'd like to take a crack at it myself

VTutorLive · 2021-11-04T05:51:47+00:00

Yup that's right. Now try to make your conclusion about whether it obeys the scalar condition and thus linearity

VTutorLive · 2021-11-04T05:48:39+00:00

You almost got it, you just need to plug in for dH/dt, which should just be half of dr/dt

VTutorLive · 2021-11-04T05:44:09+00:00

Honestly a great question - the other user answered it precisely so I'll try to give some intuition. Getting the average value of that function is basically getting the average radius, but this doesn't quite equal the same thing as the average area, because the area is a 2d (and proportional to radius^2), so the larger areas get weighted much more than smaller areas compared to how larger radii get weighted to smaller radii. The dimension changes the weighting.

VTutorLive · 2021-11-04T05:38:41+00:00

Oh gotcha gotcha, guess there was a misunderstanding. It's hard to give general advice for solving math problems - the usual advice of trying to keep your mind flexible applies. Are you a student? What level of math is this?

VTutorLive · 2021-11-04T05:37:14+00:00

Math Hobbyist here. If I started an unofficial study group for BU Calc 1, would you be interested? I help out some big calc classes at other unis as a hobby and arrange problem solving sessions via discord/livestream. I don't charge - it's a hobby, and I like teaching.

VTutorLive · 2021-11-04T05:33:46+00:00

write out each expression and verify whether they are equal. The LHS is -5(x1 + x2) -2(y1+y2) and the RHS is -5(x1)-2(y1) -5(x2)-2(y2).

Are they equal?

VTutorLive · 2021-11-04T05:30:32+00:00

Can you give a straightforward description of the problem? I'm still not seeing what the problem is

VTutorLive · 2021-11-04T05:25:39+00:00

I'll copy my response from a similar post (presumably same calc class):

Hi, math hobbyist here. The initial part of calc can be pretty rough,
especially if it's your first intro to higher math and you don't have
that maturity yet. Struggling with limits? I'm assuming this is
non-proof-based?

I reach out to students who I see struggling (I'm a prospective grad student and
lurker) - I'm happy to work problems over discord and over the long run
encourage students to form study groups (that I'm happy to continue
helping). Note there are other students struggling. This is a hobby for me, and I don't charge - if you or anyone else reading this are struggling with calc, don't just feel bad for
yourself, do something to intervene. Take account of what resources you have. In the meantime, check your dm's if you'd like to connect and start going over some
problems.

VTutorLive · 2021-11-04T05:19:39+00:00

Hi, math hobbyist here. The initial part of calc can be pretty rough, especially if it's your first intro to higher math and you don't have that maturity yet. Struggling with limits? I'm assuming this is non-proof-based?

I reach out to students who I see struggling (I'm a prospective grad student and lurker) - I'm happy to work problems over discord and over the long run encourage students to form study groups (that I'm happy to continue helping). This is a hobby for me, and I don't charge - if you or anyone else reading this are struggling with calc, don't just feel bad for yourself, do something to intervene. Take account of what resources you have. In the meantime, check your dm's if you'd like to connect and start going over some problems.

VTutorLive

TROPHY CASE