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  1. M

    Unusual set of 3 integers

    I fail to see how this is inconsistent with Mathematics in general!
  2. M

    Linear algebra: Prove that the set is a subspace

    I don't follow what you are trying to say here. I'm sorry, I must be missing something.
  3. M

    Linear algebra: Prove that the set is a subspace

    Your basis has to be a set of 3x3 matrices. That is fair, I'm just wondering the method you used. You've (assuming your calculations are correct) found what matrices in the subspace U "look like". Can you write that matrix as a linear combination of other matrices? (I haven't verified, but it...
  4. M

    Linear algebra: Prove that the set is a subspace

    What does it mean to be a basis? I don't agree with your explanation of the dimension and basis of U. Column vectors (in this case) are ##3x1## matrices. You're claiming that you can generate a whole space of 3x3 matrices with linear combinations of 3x1 matrices. The basis that you are claiming...
  5. M

    Polynomial splits over simple extension implies splitting field?

    That is a fair critique I suppose. If we wanted to be super rigorous we could fix an algebraic closure of ##\mathbb{F}## call it ##\mathbb{F}^*## with ##\mathbb{F} \subseteq \mathbb{K} \subseteq \mathbb{F}^*##. But either way, while the argument is very short, it seems like a useful lemma that...
  6. M

    Proof Practice

    You might start by trying to prove some statements about parity. Like an odd number times an odd number is always odd. Even number times anything is always even. Those are pretty straight forward. If you feel you already feel comfortable with those. Then really you should just find a good...
  7. M

    Polynomial splits over simple extension implies splitting field?

    Since ##\alpha## is a root of f and ##\mathbb{L}## is the splitting field of f. ##\mathbb{L}## must contain all roots of f, basically by definition of splitting field.
  8. M

    Collection of Lame Jokes

    What's purple and commutes? An abelian grape.
  9. M

    Separate a variable

    Three things: First, gotta have some context. We can't help you if we don't know what the question is. Second, learn latex it's not hard at all, your equations are impossible to interpret with certainty. Third, are these your equations? If not, you may quote this message and see how it was...
  10. M

    Probability - Tree Diagram problem

    Can you describe your intent with the attached tree? That is, how have you chosen your labels on your branches? Why do you know there is something wrong?
  11. M

    Proof Practice

    Also, if you scroll through the Calculus and beyond section you might find some exercises to try to prove. Or.. you might prove that there are no positive integers ##a,b,## and ##c## such that for ##n \geq 3## $$a^n + b^n = c^n$$. :-p
  12. M

    Polynomial splits over simple extension implies splitting field?

    This is a question that came about while I attempting to prove that a simple extension was a splitting field via mutual containment. This isn't actually the problem, however, it seems like the argument I'm using shouldn't be exclusive to my problem. Here is my attempt at convincing myself that...
  13. M

    Group Presentations Isomorphisms

    It's not obvious to me why simply mapping the generators to generators should define a homomorphism. Just to be sure I'm on the same page as you. Let ##a,b ## generate G, ##a',b'## generate H. Let ##\phi : G \rightarrow H## be defined by ##\phi(a)=a'## and ##\phi(b)##. From just this, It's not...
  14. M

    Group Presentations Isomorphisms

    If you are trying to show that two groups, call them H and G, are isomorphic and you know a presentation for H, is it enough to show that G has the same number of generators and that those generators have the same relations?
  15. M

    Science Jokes

    So, finite groups are virtually trivial. What's the big deal?
  16. M

    Science Jokes

    What is the contour integral around Western Europe? 0 all the poles are in Eastern Europe.
  17. M

    Proofs in Linear algebra

    I most certainly am over-thinking the assignment, that was never really in question. My point is, it seems like it would be more helpful to ask them to just calculate things than it is to ask the students to prove things, but then allow for a proof that skims the ideas without forcing them to...
  18. M

    Dimension of the span of a set of vectors

    The only condition I wanted on m is that it was not necessarily equal to n. So that it was truly an arbitrary set of vectors in ##R^n##. That way the question was just: "Given an arbitrary set of vectors, A, such that dim(span(A)) = n. Does A necessarily span ##R^n##?" This is a more concise way...
  19. M

    Dimension of the span of a set of vectors

    Ok, this is exactly what I needed to know! I've not gotten to modules yet. I'm taking a second graduate Algebra class next semester though. I'm told we will finally get into them then.
  20. M

    Dimension of the span of a set of vectors

    Yeah that second, 1 should have been an m. Not necessarily an n. (at least I don't think it should necessarily be an n).
  21. M

    Dimension of the span of a set of vectors

    My linear algebra is a bit rusty. Let ##A=\{\bar{v}_1, \dots, \bar{v}_1\}## be a set of vectors in ##R^n##. Can dim(span##(A))=n## without spanning ##R^n##? I guess I'm unclear on how to interpret the dimension of the span of a set of vectors.
  22. M

    Proofs in Linear algebra

    I'm given solutions for each homework. These solutions are generally less "rigorous" than a proof that I would turn in on any of my own homeworks. So I don't expect the proofs on the homeworks I'm grading to be something like: "Let blah be a blah such that blah.. then blah, therefor blah, and...
  23. M

    Proofs in Linear algebra

    I'm grading for a linear algebra class this semester. The class is comprised entirely of engineering majors of various flavors. The hw assigned by the professor is almost entirely "proofs" they are fairly specific proofs. Really the only thing that designates them as proofs is that the questions...
  24. M

    Schools Time off before grad school

    I'm in the middle of the stressful process of preparing for the GRE and preparing grad school applications. However, as it stands, I will have been an undergrad for 6 years after next spring when I graduate. I did an REU last summer, which was an amazing experience, but a ton of work. And I took...
  25. M

    Why is induction rigorous?

    I suppose I'm mistakingly using constructive and direct proof interchangeably. I've seen some set theory in a basic introductory proofs class, and then in my 2 undergrad Real Analysis classes, but I don't recall any novel construction of the Naturals. Do you have any suggestions as to where I...
  26. M

    Sending Information at Relativistic speeds?

    I just read this article: http://abcnews.go.com/Technology/plutos-majestic-mountains-atmospheric-haze-revealed-photo-horizons/story?id=33832751 The article itself is really cool, but something at the bottom of it caught my attention. I don't know how fast New Horizons is traveling, but say...
  27. M

    Other Should I Become a Mathematician?

    I disagree, it's not that modern mathematics has diverged from physics. It's just that there are now more fields that are interesting mathematics. PDE's are still heavily physics driven. Even other disciplines that's were thought to be strictly pure mathematics are having application is quantum...
  28. M

    Can you skip trig substitution?

    Like someone said, you never need it until you do. Once you get into DiffEq it will be very useful. Also, if there is a problem in a textbook that needs it, (there will be). Those problems are nearly impossible to do with out it. That being said, it's really not hard at all. Memorize your...
  29. M

    Why is induction rigorous?

    I understand how to construct a proof by induction. I've used it many times, for homework because it was clearly what the book wanted, but when I've tried it in a research setting, it's because I have so little control of the objects in working with. So it has become my impression that since...
  30. M

    Parity of a Permutation

    Thanks for responding, though I asked this toward the first semester of Abstract Algebra, I just took the final for the second semester of abstract algebra. I had this figured out at this point. But hopefully someone else finds this useful.
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