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...
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...
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...
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...
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.
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...
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?
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
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...
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...
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?
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...
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...
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.
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.
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...
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...
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...
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...
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...
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...
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...
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...
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.