Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...
0
votes
0answers
25 views
Kernel and image of a set of linear maps [duplicate]
I'm having some trouble with this question (I think I didn't reach an acceptable answer - also hope I used the correct terminology).
Let $V,W$ be sets of vectors over $F$ of finite dimensions. We'll ...
0
votes
1answer
25 views
Projection transormation, proof of existence and uniqueness
Question:
A Projection Transformation is defined to be a linear transformation from V to V that satisfies $T^2=T$.
Let $V=U \oplus W$ . Show that there exists only one linear projection $T:V \to V$ ...
2
votes
1answer
50 views
Finding a kernel of a linear transformation of linear transformations.
Question:
Let V,W be vector spaces over field F.
We mark L(V,W) as the vector space of linear transformations from V to W.
Let $v_0 \ne 0$. We define a transformation: $\Psi: L(V,W) \to W$ that sends ...
1
vote
1answer
44 views
What about this $\lim_{x \to \infty}\frac{3x+4}{\sqrt[5]{x^9+3x^4+1}}$?
When I saw this limit, I didn't even try to solve it by an algebraic method. I thought about the assyntotic concept.
In the example,
$$\frac{3x+4}{\sqrt[5]{x^9+3x^4+1}}\sim ...
0
votes
1answer
43 views
ordering of a group
An ordering of a group $G$ is a linear ordering $<$ on (the underlying set of) $G$
that satisfies, in addition,
$a < b \implies ac < bc ∧ ca < cb$,
for all $a,b, c \in G$.
Show that a ...
1
vote
2answers
30 views
Checking independence of random variables.
I'm revisiting the coupon collector's problem and I'm not sure how to prove that my variables are independent. Here's what I have:
Let $X$ denote the number of tries required to collect all the ...
4
votes
1answer
52 views
Counterexamples in Double Integral
I need to:
$a.$ Give an example of function $f:\mathbb{R\times R}$ $\to$
$\mathbb{R}$ with domain in $[0,1]^2$ so that double integral exists but the function is not Riemann integrable.
$b.$ Give ...
1
vote
1answer
29 views
Maximal consistent set. Decomposition lemma
Let $\Gamma$ be a maximal consistent set. Prove: $\varphi \lor \psi
\in \Gamma \iff \varphi \in \Gamma $ or $ \psi \in \Gamma$.
Now define $V_{\Gamma}: Q \to \{ 0, 1 \}$ as follows:
...
0
votes
3answers
45 views
If the union of $A$ and $B$ is linearly independent then the intersection of the spans $= \{0\}$
$\newcommand{\sp}{\operatorname{sp}}$ Let $V$ be a vector space over $F$ field, and let $A,B$ be two different, disjoint, non-empty sets of vectors from $V$.
Prove or disprove the following:
...
4
votes
1answer
30 views
Can I prove continuity of a function of two variables in this way?
Common approach in handling functions of two variables is to express this function in polar coordinate system. For example, in the classic example $$f(x,y)=\left\{\begin{array}{lr}\frac{xy}{x^2+y^2} ...
10
votes
1answer
99 views
Functoriality of the Fundamental group
The fundamental group is a functor from the category of pointed topological spaces to the category of groups.
Therefore every base-point preserving continuous function $f$ between pointed ...
6
votes
1answer
66 views
Finding the antiderivative of $\frac 1{(1-x^m)^n}$,with $n,m\in\Bbb N$
For $m,n \in \Bbb N$, find the antiderivative of $g:(0,1)\rightarrow\mathbb{R}$ defined by:
$$g(x)=\frac{1}{(1-x^m)^n}$$
Mathematica gives a result with functions we didn't learn about yet. The ...
2
votes
1answer
32 views
Evaluating Complex Line Integrals
Calculate $\int_{\gamma}\frac{\Re(z)}{z-\frac{1}{2}}dz$ and $\int_{\gamma}\frac{\Im(z)}{z-\frac{1}{2}}dz$ when $\gamma$: $|z|=1$ is positively oriented.
This is what I have tried to do, starting ...
1
vote
0answers
43 views
$\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)}$
I'm stuck on my last exercise. Could you help?
$$\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)} \ dx$$
2
votes
4answers
59 views
Approximation of alternating series $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + …$
$\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$
I am asked to find the no. of terms needed to approximate the partial sum to be within 0.0000001 from the convergent ...
1
vote
3answers
38 views
Calculate two vectors given their norms and angle
For two vectors $\mathbf{u,v}$ in $\mathbb{R}^n$ euclidean space, given:
$\|\mathbf{u}\| = 3$
$\|\mathbf{v}\| = 5$
$\angle (\mathbf{u,v})=\frac{2\pi}{3}$
Calculate the length of ...
0
votes
1answer
13 views
Geometric sequence, finding the first term using only the sum, the number of terms and value of one term.
In Geometric series: S = 56, a(2) = 16 and n = 3
S - sum, a(2) - second term, n - number of terms
Is it possible to get a(2) and a(3) from here? (If yes, hints would be awesome)
Thank You!
-1
votes
1answer
88 views
How to solve this equation for $r$? [closed]
I have a problem....
I have to express unknown "r" from from this equation:
$$Y\times r=(-Z\times 2r\times L)+(z\times K^2\times 0.5L)$$
Can someone help me ?
1
vote
3answers
88 views
$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?
Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities:
$$\|f*g\|_q\leq ...
0
votes
0answers
78 views
RSA: What message will Alice receive?
In RSA, Alice chooses $p=47$, $q=57$, public key ($n=2679$, $e=11$). When Bob sends the message $m=3$, what is the message that Alice will read?
2
votes
3answers
97 views
If $\operatorname{sp}(A) \cup \operatorname{sp}(B)=\operatorname{sp}(A\cup B) \Rightarrow A\cup B$ is linearly dependent
$\newcommand{\sp}{\operatorname{sp}}$
Let $V$ be a vector space over $F$ field, and let $A,B$ be two different, disjoint, non empty sets of vectors from $V$.
If $\sp(A) \cup \sp(B)=\sp(A\cup B) ...
0
votes
1answer
21 views
Linear interpolation of points in isometric isomorphic spaces
Suppose that we have two spaces $\mathcal{F}$ and $\mathcal{H}$ and we know that $\mathcal{H}$ is isometric isomorphic to $\mathcal{F}$, so that distances and angles are preserved. Note that we are ...
0
votes
0answers
29 views
How to prove the global dimension of the polynomial ring $F[x_1,…,x_n]$ is $n$?
I am trying to prove that the global dimension of the polynomial ring $F[x_1,...,x_n]$, where $F$ is a field , is exactly $n$. And by Koszul Complex, I know its global dimension is greater than or ...
1
vote
0answers
32 views
A strange characterisation of cyclic groups [duplicate]
"A finite group is cyclic if, for any integer m, the number of elements
of order dividing m is at most m."
I have never seen this characterisation of cyclic groups before. How do I prove this? I hope ...
0
votes
2answers
30 views
Whether an infinite series can be tested by integral test
I am asked whether the following infinite series can be proved to be convergent by integral test.
$$\sum_{n=1}^\infty n e^{6 n}$$
so I integrate it
$$\int_1^{\infty}\ n e^{6n}\, dn$$
and find it ...
0
votes
0answers
33 views
turne into dimension one
who can help me to turn this problem into a problem on dimension one ?
how to write (1,1) and (1,2) in dimension one ?
Please ,help me
Thank you
0
votes
1answer
24 views
Euler's method for second order differential equation
Not really homework but sample exam.
The question is to use Euler's Method to approximate Y:
$Y''(t) = Y'(t) - 2Y(t)$
$Y'(0) = Y(0) = 1$
with $t_0 = 0$ and $h=0.2$
So what I did:
First ...
3
votes
4answers
62 views
Prove or disprove the following statements involving greatest common divisor
Help with prove or disproving either of these statements would be really appreciated, one or the other is fine, I just need a start or a solution to one and I'm sure I could probably figure the other ...
3
votes
1answer
31 views
Calculation of ordered pair $(x,y,z)$ in $x^2 = yz\;\;,y^2=zx\;\;,z^2 = xy$
(1) Total no. of integer ordered pair $(x,y,z)$ in $x^2 = yz\;\;,y^2=zx\;\;,z^2 = xy$
(2) Total no. of integer ordered pair $(x,y,z)$ in $x+yz = 1\;\;,y+zx = 1\;\;,z+xy = 1$
My Try:: (1) Clearly $ x ...
2
votes
1answer
30 views
Finite Extensions and Roots of Unity
Two questions; the hint I've been provided is that they are, in fact, related.
Prove that a finite extension of $\mathbb{Q}$ contains finitely many roots of unity.
What is the largest (finite) ...
2
votes
1answer
23 views
Basic independent probability question
This question is a homework question.
The question states:
An airline can seat 100 people. Historically, the airline has noticed that each customer shows up independently and with probability ...
1
vote
4answers
74 views
What is the derivative of $\ln(4^x)$?
What is the derivative of $\ln(4^x)$ (which I believe is also equal to $x\ln4$)?
Is it $\dfrac{1}{x\ln4}$?
1
vote
2answers
37 views
Values of a parameter $x$ in an infinite series that makes it converge
I am required to find the values of $x$ in the following infinite series, which cause the series to converge.
$$\sum_{n=1}^\infty \frac{x^n}{\ln(n+1)}$$
I tried to use the ratio test, and found that ...
2
votes
2answers
17 views
Relationships of Eigenvalues in Algebraic Closure
Suppose that $k$ is a field, and $A \in M_n(k)$ is a matrix that becomes diagonalizable over $\overline{k}$, the algebraic closure of $k$. Let $\lambda_1, \ldots, \lambda_n$ denote the (not ...
3
votes
1answer
38 views
Finite Projective Dimension implies non vanishing Ext
Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$?
Can't we write the free module as a direct ...
1
vote
2answers
24 views
Why does $7^{2\ln x}\cdot \ln(7) \cdot (2/x)$ equal to $7^{2\ln x}\cdot \ln(49) /x$?
While reviewing, I came upon this problem which has the derivative
$7^{2\ln x}\cdot \ln(7) \cdot (2/x)$
simplified to
$7^{2\ln x}\cdot \ln(49) /x$
How/why is it simplified like that?
5
votes
1answer
38 views
$\inf A = -\sup(-A)$
Let $A$ be a nonempty subset of real numbers which is bounded below. Let $-A$ be the set of of all numbers $-x$, where $x$ is in $A$. Prove that $\inf A = -\sup(-A)$
So far this is what i have
...
2
votes
2answers
61 views
How do I solve for $dy/dx$ if $y=\ln (\sin x+\ln x)$?
Solve for $\frac{dy}{dx}$ if $y=\ln(\sin x+\ln x)$.
I know how to solve for integrals involving $du$ and $u$, but how do I do this type of problem (I think it's the opposite of the integral problem)?
...
-2
votes
1answer
33 views
Why is this expectation calculation wrong?
Let
$E(X|A)=2+E(X)$
$E(X|B)=3+E(X)$
$E(X)=0.5E(X|A)+0.3E(X|B)$
Now to find $E(X^2)$ I did
$E(X^2|A)=4+E(X^2)$
$E(X^2|B)=9+E(X^2)$
$E(X^2)=0.5E(X^2|A)+0.3E(X^2|B)$
What did I do wrong here?
...
0
votes
2answers
44 views
Prove there is a tree with $n$ vertices having degrees $d_1, d_2…d_n$
For $n ≥ 2$ suppose $d_1, d_2,....d_n$ are positive integers with sum $2n - 2$. Prove there is a tree with n vertices having degrees $d_1, d_2....d_n$. I'm at a loss on this one. I'm sure it's pretty ...
1
vote
0answers
34 views
I'm having trouble finding this matrix $T$ relative to $\mathcal B$ and the standard basis $\mathcal E$ for $\mathbb R^2$
This was a homework assignment, but unfortunately it was the last homework assignment of the semester so I never got feedback and I'm just reviewing it for a final. I'm supposed to let $\mathcal ...
2
votes
2answers
36 views
Confused about combinatorials
How do I solve 4$\cdot$6 = 8$\cdot$3 by a combinatorial proof? How can I start this proof? I know that I can show a two pictures that represent 24 but I'm not entirely sure how to go about this. ...
2
votes
1answer
31 views
How to prove Chebyshev–Gauss quadrature integrate polynomial of degree less than $2n-1$ exactly
What I want to ask is mentioned in the title.
For example: how can we show that ...
2
votes
0answers
49 views
Double Integral Homework Problem
Here's the problem statement of the question which I am stuck on:
Let $R_{1}$ denote the rectangle $[0, 5] \times [-4, 4]$, $R_{2}$ the rectangle $[0, 5] \times [0, 4]$, and $R_{3}$ the rectangle ...
2
votes
2answers
31 views
Quadrature formula
How can we find a quadrature formula $\int_{-1}^1 f(x) dx=c \displaystyle \sum_{i=0}^{2}f(x_i)$ that is exact for all quadratic polynomials?
Thanks for help.
0
votes
2answers
71 views
Show using induction (coupled linear recurrences)
Some homework help would be greatly appreciated, took a screenshot and made an image to make it easier to show and get help with.
(2) Consider the numbers defined recursively by $a_1=3$, $c_1=5$, ...
4
votes
2answers
58 views
Positive semidefiniteness of a block matrix of positive semidefinite matrices
Given any symmetric matrix $\mathbf{M} = \begin{pmatrix}
\mathbf{A} & \mathbf{B}\\
\mathbf{B}^\mathrm{T}& \mathbf{C}
\end{pmatrix}$, the following conditions are equivalent:
(1) ...
2
votes
0answers
25 views
Non-isomorphic simple extensions of the same degree of a field of positive characteristic
Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic.
I thought of an example where they are ...
1
vote
0answers
20 views
Gaussian quadrature with arbitrary weight function
In class, our professor told us how to evaluate the integral $\int_a^bw(x)f(x) dx$ by finding the Gaussian nodes $x_i$ and weight $w_i$ with weight function $w(x)=1$ (also known as Legendre ...
4
votes
2answers
61 views
Cardinality of $GL_n(K)$ when $K$ is finite
I don't know how to do the last task of an exercise.
Let $K$ be a field, $G=GL_n(K)$ and $X=K^n\backslash\{0\}$.
First task: Show that $G \times X \to X$, $(A,x)\mapsto Ax$ defines an action of $G$ ...
