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Dec
20
comment Group of $r$ people at least three people have the same birthday?
Consider the following scenario: persons A and B have birthday X while persons C and D have birthday Y. This sort of situation is not accounted for if you're only counting ways there can be precisely one pair of persons sharing a birthday. It is possible for multiple pairs of persons to share birthdays without any three sharing a birthday.
Dec
20
comment Integral Property: $\int^a_{0}f(x)dx=\int^a_{0}f(a-x)dx$ [Proof by definition of Riemann Sums]
Proof by Riemann sums: $a+b+c+\cdots+z=z+y+x+\cdots+a$.
Dec
19
comment Roots of a degree $3$ polynomial with real coefficients.
See if you can solve for the polynomial with linear algebra. Also seems like you didn't write out the full question. Shouldn't there be some kind of "which of the following are possible?" thing written in the problem statement?
Dec
17
comment Finding an example of a set $G$ which is not a group
The class of binary operations which yield group structures is infinitessimal among all binary operations. All you'd need to do is take a random stab in the dark and you'd surely hit one that's not a group structure. In fact you already did: the set of all square matrices under multiplication does not form a group: there are noninvertible matrices.
Dec
16
comment Collection of smooth real valued functions on smooth manifold has ring structure.
Do you know how to add and multiply functions? Do you know what we do with functions and points? Please say what thoughts you have.
Dec
15
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Dec
15
comment Good explanation of $\mathbb{R}^3\rightarrow\mathbb{R}^2$
Two ... outputs, to finish the sentence. Of course an alternative interpretation is that a 3-tuple counts as a single input and a 2-tuple counts as a single output.
Dec
15
comment Orthogonal Projection - property of an orthogonal operator or something that needs to be proven?
I will explain some of the geometry using my newest account. (Also, no $P$ is not an orthogonal transformation, but it is an orthogonal projection. With any other kind of projection there will be an $x$ for which $\|Px\|>\|x\|$.)
Dec
14
comment Orthogonal Projection - property of an orthogonal operator or something that needs to be proven?
What happens is $\langle u+u',u\rangle=\langle u,u\rangle+\langle u',u\rangle=\langle u,u\rangle$. The key ideas here are: the inner product is linear in both arguments, it evaluates to 0 if (and only if) its two arguments are orthogonal, and the idea of orthogonal projections (every vector can be decomposed into parallel and perpendicular components with respect to some subspace).
Dec
14
comment Orthogonal Projection - property of an orthogonal operator or something that needs to be proven?
There is an orthogonal decomposition $X=U\oplus U^\perp$. So we can write $x=u+u'$ with $u=Px$ and $u'$ orthogonal. Then what happens with $\langle u+u',u\rangle$ (that's $\langle x,Px\rangle$)?
Dec
14
comment Orthogonal Projection - property of an orthogonal operator or something that needs to be proven?
Geometrically, why would you say $\| Px\|\le \|x\|$ is true? Because $Px$ is the base and $x$ the hypotenuse in some right triangle. You can work with that to write down a proof. (Also, do you understand the geometric reasoning behind $\langle Px,Px\rangle=\langle x,Px\rangle$?)
Dec
10
comment What is $\mathbb{S}^{1}/\{\pm {1},\pm {i}\}$ isomorphic to
Those are four different points, four different complex numbers.
Dec
10
comment Left invariant Vector Field on $S^2$
You mean the sphere isn't a Lie group. What do you think invariance means?
Dec
6
comment What do we know on the number of groups of a given order?
Another item of note is that product is an upper bound for the number of group structures on a set of size $n$, which is itself bigger than the number of isomorphism classes of groups of size $n$.
Dec
4
comment What's the exact definition of polynomial in $\mathbb{Z}_n$?
Polynomials are symbolic expressions, polynomial functions are actual functions. Two different polynomials can represent the same polynomial function. In particular, $x^p-x$ and $2(x^p-x)$ and $0$ are all the same function on $\Bbb Z/p\Bbb Z$, but are different polynomial expressions. It is a fact that over any field, if $f(x)$ has distinct roots $r_1,\cdots,r_n$, degree $n$ and leading coefficient $a$, then it equals $a(x-r_1)\cdots(x-r_n)$ not just as functions, but as polynomial expressions. Also, Euler's says $x^p\equiv x$, not $x^p\equiv -x$.
Dec
4
comment Ring $x^2=x$ then $2x=0$
If you can square elements then you can square algebraic expressions.