@MaryStar we know that the kerB is equal in size to A, but in general no consider the map B maps u to 0 iff u is in image of A else B maps to litter-ally anything except 0
@MaryStar Not usually, in certain instances we can for instance if there exist only the 0 vector or the space your working in is the same size as ImA, The main point is B can be strictly larger than the image of A
now i can actually see it, i didn't know how to render latex before like 5 comments ago. try leaving the omegas in and take the dt to the outside. I'm not to comfortable with what a sin of a infinitesimal but i think it would mean the same thing?
Interesting notation that came up in that lecture: $\displaystyle\underbrace{V\otimes V\otimes\dotsb\otimes V}_{d~V\rm s}$ got written as $V^{\otimes d}$
like if you spent 2 years working with a particular technique, what are you going to do when confronted with a new problem? you will try to apply that technique even if it cannot possibly work
yes it can be useful, but it can also lead you astray
That hammer-nail quote would probably be why I'm watching a video about Hopf algebras. No idea how they're used (and only a vague idea of what they are), but it's good to have another tool under my belt
Well, no, the reason I started watching it was that I was bored :P
@ForeverMozart i never said it was easy, i said it was hard. and yeah but solving the three Diophantine equations generated by that method is nontrivial.
For the triangle that is .from the unit square its known that it cant exist on the diagonals or the side of the square. its also known that if there exist a point there exist 8 points due to symmetries and if a point exist its a rational point in the traditional sense. little else is known
Can someone explain to me what Euler's formula used for? Or is it just a random relationship between e^ix and trig functions (complete newb here only took lower division math)
Like what can i write about it if i don't know complex analysis other can it's used in complex analysis? Any dumbed down version of what I could put that everyone would understand?
because i is in the equation in the first place its imaginary but its results are applicable in the real plane whenever one gets real values
and no unfortunetly i dont have access to volume
maybe its better stated as all real numbers are complex numbers. but not all complex are reals. so it has applications in the reals but its not limited to implications in either field
^complexes are almost vectors. and sort of complex numbers are an extension of real numbers over the reals as a field meaning that all complex numbers are of the form a+bi for a,b in reals. that being said all real numbers are complex numbers along the line b=0
you may not be ready for this mind blowing so sit down
4^(.5) = 2,-2 in the same way (-1)^.5=i,-i. This is why a complex numbers always come in pairs, and you can do what i did above replacing (i with -i) or replace (cos(x) with sin(x)) any combinations of these will show the same pattern due to the fundamental relationship between them
i.e. if a polynomial has a root a+bi then it must also have a root a-bi
Let me butt in for a sec. Here's why the connection to cos/sin works: e^{ix} = cos(x) + i*sin(x). Differentiate both sides n times, you get i^n * e^{ix} on the left. Equate real and imaginary parts on the right, and you'll see that the alternating pattern of derivatives of cos and sin comes right from the pattern of the powers of i. Although it's more fun to think of this stuff in geometric terms...
Exactly. Conveniently, most of the rules you learned for differentiating real functions still apply for differentiating complex functions. Just treat a complex constant (like $i$ or $3i + 4$) as a constant.
in truth i never truly understood complex numbers until first semester abstract algebra. This class will show you more of the fundamentals about the complex field and its behavior. and answer just what are these field things anyway
see if it were Q, anything nonzero would divide anything else, making the problem trivial: every positive number y would be a solution @user19405892 @anon
not right now but i'm not permanetly not. I.e. i don't have my Ph.D but i'm an all but dissertation. so they allow me to be an assistant with worse pay and double the work.
hey mr combinatorics @EricStucky how would i go about proving that in the long term f(x) = (3x+1)/2 if x odd x/2 if even, repeat until x=1. How would one go about proving that in the long term both steps are equally likely
