Questions tagged [solution-verification]
For posts looking for feedback or verification of a proposed solution. This should not be the only tag for a question.
33,020
questions
-1
votes
0answers
6 views
Show that if A is an m x n matrix having linearly independent columns, then the entries of the reduced echelon form of A consist of 1s and 0s.
I got the following question on an exam -
Show that if A is an m x n matrix having linearly independent columns,
then the entries of the reduced echelon form of A consist of 1s and
0s.
Now looking ...
0
votes
3answers
38 views
want to check a proof i wrote
i'm taking a non-credit real analysis course online and this is a proof i wrote for one of my answers in my first problem set.
my solution is different from the instructor's, but he said in the video ...
-1
votes
1answer
30 views
how to give elements of $A × B$ and $A ∩ B$
$A$ = {$3n+1: n∈Z$} and $B$ = {$12m+7: m∈Z$}
I have proven that $A⊄B$ by showing that $3n+1$ $⇒$ $3n+1+7-7$ ⇒ $3n-6+7$ ⇒ $3(n-2)+7$ ⇒ $4((3(n-2))+7)/4$ ⇒ $12(n-2)(1/4) +7$ ⇒ $12m-7$, where $m = (n-2)(...
0
votes
2answers
15 views
Surjective linear transformation of finitely generated free modules and their determinants
I kind of got stuck on Aluffi Algebra Exercise 6.4 of chapter 6 which is:
Let $F$ be a finitely generated free $R$-module and let $\alpha$ be a linear transformation of $F$. Give an example of an ...
2
votes
1answer
35 views
Exercise 2.6 (General form of the Chinese remainder theorem) from Eisenbud's Commutative Algebra
Exercise 2.6 (General form of the Chinese remainder theorem): Let $R$ be a ring, and let $Q_1,\dots,Q_n$ be ideals of $R$ such that $Q_i+Q_j = R$ for all $i\neq j$. Show that $R/(\bigcap_i Q_i) \cong \...
3
votes
1answer
57 views
Find all real numbers $x$ such that $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number
I want to find all values of $x\in \mathbb R$ such that the value of $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number.
I solved it as follows:
$x+2\sqrt{x}-1\ge 0$
$(\sqrt{x}+1)^2-2\ge 0$
$...
0
votes
0answers
20 views
Given a function $f:N\rightarrow N$ it is true that $f(1)=43$, $f(2)=142$ and $f(n+1)=3f(n)+f(n-1)$, $n\ge 2$. Prove that $(f(n+1), f(n))=1$
Given a function $f:N\rightarrow N$ it is true that $f(1)=43$, $f(2)=142$ and $f(n+1)=3f(n)+f(n-1)$, $n\ge 2$. Prove that $\frac{f(n+1)}{f(n)}$ is irreducible.
I proved it as follows:
If $n=1$ it ...
1
vote
0answers
27 views
Subcomplex is closed
Hatcher, p. 520 in the appendix:
A subcomplex of a CW complex $X$ is subspace $A \subset X$ which is a union of cells of $X$ such that the closure of each cell in $A$ is contained in $A$ ... It is ...
0
votes
0answers
21 views
I am stuck with integration and differentiation that implies continuity
I am not sure about continuity of a differentiable function after integration.
Let's assume, $f$ is differentiable at $[0, \infty)$
$$\int f(x)dx = F(x)$$
Where $f$ is the derivative of $F$
Now the ...
-1
votes
1answer
38 views
$X$th Permutations of $1,2,3,4,5$ is $25314$. Find X
The permutations of $1,2,3,4,5$ are lexicographically ordered.
$X$th permutation is $25314$. Find $X$.
I am getting $1*4! + 3*3! + 1*2! + 1=45$. Is it correct?
Reasoning:
There are $1*4!$ numbers of ...
1
vote
0answers
46 views
A possible solution for a specific case of the Beal's Conjecture?
According to wikipedia's page on Beal's Conjecture, it states: $A^x + B^y = C^z$, where $A, B, C, x, y$ and $z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $A, B$ and $C$ ...
0
votes
0answers
18 views
Doubt in Continuity chapter of Rudin(Question - 17)
Let $f$ be real function defined on $(a,b)$. Prove that the set of points at which $f$ has a simple discontinuity is at most countable.
Proof: Let $E$ be the set of points at which $f$ has a ...
0
votes
1answer
11 views
Condensed SVD decomposition of an outer product
Let $A = uv^{T} \in \mathbb{R}^{m \times n}$. Find the (condensed) SVD decomposition of $A$.
Theorem (Condensed SVD decomposition)
Let $A \in \mathbb{R}^{n \times m}$ be a non-zero matrix of rank $r$....
-1
votes
2answers
30 views
Does this proof make sense(The well-ordering principle for N)?
I need to explain the proof given by Des McHale in his "The well-ordering principle for N". I was hoping you could tell me if it makes sense and what else I could add to it. This is what I ...
1
vote
1answer
39 views
Proof by counterexample
If $n$ is prime, then $\sqrt{n}$ is irrational. Prove this statement.
If I were to prove this using proof by contradiction, I would do:
Suppose $n$ is prime and $\sqrt{n}$ is rational. Let $\sqrt{n}=\...
0
votes
0answers
14 views
Show the columns of a lower triangular matrix are linearly independent if and only if the diagonal entries are non-zero.
I should note that I can't use determinants for this proof. This is what I came up with:
Let $A$ be a lower triangular matrix and suppose $A^1, ..., A^n$ are linearly independent. Clearly if the ...
3
votes
1answer
49 views
Let $(x_n)$ and $(y_n)$ be sequences such that $(x_n)$ is a subsequence of $(y_n)$ and $(y_n)$ is a subsequence of $(x_n)$. $x_n=y_n$ for all $n$?
Let $(x_n)$ and $(y_n)$ be sequences such that $(x_n)$ is a subsequence of $(y_n)$ and
$(y_n)$ is a subsequence of $(x_n)$. Given that $x_n$ converges, does it follow that $x_n=y_n$ for all $n$?
I ...
1
vote
1answer
37 views
Conjecture: $(\frac{a}{c})^2+(\frac{b}{d})^2=e^2$ has no solutions in distinct positive integers if the fractions are fully simplified non-integers
Given that $a,b,c,d,e,f$ are all distinct positive integers, prove that there are no solutions for the following knowing their fractions are non-integers and completely simplified:
$$\left(\frac{a}{c}\...
0
votes
2answers
25 views
Truth value of statements about empty set
I have a problem:
Let P be the statement " x $\in$ A and x $\in$ $\mathbb{Z}$ "
Determine the truth value of statement: ($\forall$x)P $\Longrightarrow$ ($\exists$x)P
Is there a set A for ...
1
vote
1answer
32 views
What we need to show, $ab=1\bmod k$ or $(ab)\bmod n=1\bmod k$? For proving $U_k(n)≤U(n)$
I need to show that,
For each divisor $k$ of $n$, $U_k(n)$ is subgroup of $U(n)$ where, $U_k(n)=\{x\in U(n) : x=1\bmod k\}$
My attempt: as $U(n)$ is finite group for each $n\in\mathbb{Z}^+$.
Hence ...
0
votes
0answers
30 views
System of equations wrong solution $(x, y, z) = (3, 3, 2)$
Find the real solutions for the following system of 2 equations:
$z^2 - 4z = -9 - 2xy$
$x(x - 6 - y) + y(y - 6 - x) = -13$ (A book of problems for 8th graders)
The book says that $(x, y, z) = (3, 3,...
2
votes
0answers
69 views
Finding convergent subsequence of $(\sin n)$ in $\mathbb R$
Convergent subsequence of $(\sin n)$ exists by Bolzanno Weierstrass theorem. I am trying to construct a convergent subsequence of the sequence $(\sin n)$.
My attempt: I note that the sequence $(x_n)$,...
0
votes
0answers
25 views
Deriving the definition of a simplex
This posted emerged out of the following question posted in "Introduction to Linear Algebra by Gilbert Strang (5th edition)":
:
Figure $1.5$b is given as follows:
"$c+d+e \leq 1$" ...
1
vote
1answer
37 views
Set Theory element argument proof help please
Let $S$ be any finite set and suppose $x \notin S$. Let $K=S \cup \{x\}$. $X=\{T \subseteq K : x \in T\}$.
Prove that every element of $X$ is the union of a subset of $S$ with $\{x\}$, and that if ...
2
votes
3answers
51 views
Unique questions about pentagon
Consider the (non-regular) pentagon with consecutive vertices at (-1,-1), (-1,1), (0,2), (1,1), and (1,-1).
a) Prove that there is no circle that is tangent to all 5 sides of the pentagon
b) Is there ...
2
votes
1answer
20 views
Showing that a set is open with metric
Let $(A, d_1)$ and be a metric. Suppose $d_2: A \times A \rightarrow \mathbb{R}$ be defined in this way:
For $a, b \in A$, $d_2(a, b) = \frac{d_1(a, b)}{1 + d_1(a, b)}$.
Show that $E \subset A$ is ...
1
vote
0answers
25 views
Supremum of the sum of two subsets equals to the sum of two suprema
Let $A$ and $B$ be upper bounded subsets, and let $A+B$ be the subset of numbers $x+y$ with $x \in A$ and $y \in B$. Prove that $\sup(A+B)=\sup(A)+ \sup(B)$.
Let $a =\sup A$ and $b =\sup B$, it is ...
2
votes
2answers
67 views
I need a limit definition for the Hessian, does this work?
Let $f: \mathbb{R}^n \to \mathbb{R}$ be of class $C^2$. Let $x$ be a non-degenerate critical point of $f$. Prove that there is an open neighborhood of $x$ which contains no other critical points of $f$...
3
votes
1answer
63 views
Eisenbud Exercise 2.4 (Practice with $\text{Hom}$ and $\otimes$)
I finished doing Exercise 2.4 from Eisenbud: Commutative Algebra, with a View Toward Algebraic Geometry, and I was wondering if I could have someone look over my solutions to check if everything is ...
0
votes
1answer
28 views
What's wrong with my Surface Area of a solid of revolution formula?
When I learnt about the derivation for the formula $$V=\pi\int_{x_1}^{x_2} y^2~dx$$
where $V$ is volume of the solid generated when $y=f(x)$ is rotated about the $x$ axis by $2\pi$ radians between $...
0
votes
1answer
27 views
Is cocountable topology generally separable.What else is wrong with this table?
Is the cocountable complement topology always separable? Also are there Really any other issues with this table I created?
$$\begin{array}{cc|c}
& \mbox{T1} & \mbox{Hausdorff}&\mbox{...
4
votes
1answer
52 views
Which $N$ are multiples of $2020$ with all distinct digits where swapping any two of them makes $N$ not a multiple of $2020$?
The question
I have attempted problem 3 from the Senior O-level Tournament of Towns paper, Fall 2020. The question is as follows:
A positive integer number N is divisible by 2020. All its digits are
...
4
votes
5answers
95 views
Avoiding brute force: determining when a specific polynomial in $\mathbb{Q}[x]$ is an integer for any integer $x$
I have to prove that $\frac{1}{5}n^5+\frac{1}{3}n^3+\frac{7}{15}n$ is an integer for any $n$. I solved this by brute-force, exhausting all the possibilities methods. I was wondering if there was a way ...
2
votes
2answers
36 views
Counting problem assigning student to groups
Say I want to assign $4$ students $a,b,c,d$, to three teams, and each team needs at least one student in it. Only one issue, $a$ and $b$ cannot be in the same team. How many ways are there to assign ...
0
votes
2answers
26 views
Counting Questions about Position
Lets say we have a student organization of 8000 students. 5000 are undergrads and 3000 are master students. There is an organization that has a president position, and three VP positions.
a) How many ...
1
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0answers
45 views
Proving the Lebesgue outer of a set is equal to its volume
I want to prove the following statement
if $A$ has volume, then
$$
\left( \operatorname{vol}(A) = \int_A \chi_A \right) = \inf\left\{ \sum_{i=1}^\infty \operatorname{vol}(S_i) \ | S_1, S_2,\cdots \...
0
votes
0answers
14 views
OCR A Level Projectile motion question: Solution verification
Question taken from the book is in the image above.
$$$$
The solution in the back of the book is Solution $2$. However, I believe that the partcile is on the way down when $\theta = 37^\circ.$
$\tan \...
0
votes
3answers
23 views
Let $f:(0,\infty)\to \mathbf{R}$ and $L,L' \in \mathbf{R}$. If $\lim_{x\to\infty}f(x)=L$ and $\lim_{x\to\infty}f'(x)=L'$, show that $L'=0$
Let $f:(0,\infty)\to \mathbf{R}$ be differentiable on $(0,\infty)$ and $L,L' \in \mathbf{R}$. Suppose that $\lim_{x\to\infty}f(x)=L$ and $\lim_{x\to\infty}f'(x)=L'$. Prove that $L'=0$. I know that ...
1
vote
1answer
17 views
Prove basis for product topology is collection of all sets with finitely many factors restricted to open sets
Prove a basis for the product topology on $\prod_{a \in \lambda}X_{a}$ is the collection of all sets of the form $\prod_{a \in \lambda}U_a$ where $U_a$ is open in $X_a$ for each $a$ and $U_a=X_a$ for ...
4
votes
1answer
64 views
Math Olympiad question: Quadratic Equations.
Math Olympiad question:
If the equation in $x$ has real roots, then find the value of $a$ and $b$.
$x^2 + 2(1 + a)x + (3a^2 + 4ab + 4b^2 + 2) = 0$
Approach:
For at least one real root:
$$b^2 -4ac \...
0
votes
1answer
52 views
If $\lim_{x\to \infty} f(x) = c\neq 0$ and $\lim_{x\to \infty} g(x) = 0$, then $\lim_{x\to \infty} \frac{f(x)}{g(x)}$ does not exist.
This is just a minor part of another problem and intuitively clear, but I think that it is necessary to have a proof of this assertion.
If $\lim_{x\to \infty} f(x) = c\neq 0$ and $\lim_{x\to \infty} ...
3
votes
2answers
40 views
Prove that if $\lim_{n\to\infty} \frac{x_n}{y_n} = 0$, then $\lim_{n\to\infty} x_n \div \frac{x_n + y_n}{2} = 0$
Let $\{x_n\}$ and $\{y_n\}$ be positive sequences. Assume $\lim_{n\to\infty} \frac{x_n}{y_n} = 0$. I have to prove if the claim $\lim_{n\to\infty} x_n \div \frac{x_n + y_n}{2} = 0$ is always true, ...
1
vote
0answers
13 views
Does Edwards need to explicitly state $a\in\mathcal{W}$ in his Inverse Mapping Theorem?
In view of the answer to my question: A function is continuously differentiable in an open neighborhood of $x_o$ and differentiable at $x_o$, is it continuously differentiable at $x_o$? it seems to me ...
1
vote
1answer
49 views
Find $f_{xy}(0,0)$ and $f_{yx}(0,0)$ for the following function
Find $f_{xy}(0,0)$ and $f_{yx}(0,0)$ for the following function:
$$f(x,y) =
\begin{cases}
\frac{xy}{x^{2}+y^{2}} & (x,y) \ne (0,0) \\
0 & (x,y) = (0,0)
...
7
votes
1answer
83 views
$\lim_{x\to\infty}\frac{1-\cos x}{1-\sin x}$
Evaluate
$$S=\lim_{x\to\infty}\frac{1-\cos x}{1-\sin x}$$
Its quite simple to conclude that this limit does not exist.
Now here comes the interesting part. Consider the limit
$$L=\lim_{x\to\infty}\...
0
votes
2answers
34 views
Proving an inverse function is continuous
Let $f: (a,b) \to \mathbb{R}$ be a continuous function with $f'(x) \neq 0$ for all $x$. Prove that $g: f(a,b) \to (a,b)$ is continuous.
Here is my attempt.
Let $\epsilon > 0$, $x \in (a,b)$. and ...
0
votes
0answers
9 views
Unipotent upper triangular matrices with integer entries is Zariski dense
Let $N$ be the group of matrices $\begin{bmatrix} 1 & z \\ 0 & 1 \end{bmatrix}$ for $z \in \mathbb{C}$, let $\Gamma$ be the subgroup of $N$ with $z \in \mathbb{Z}$.
I wish to show that $\Gamma$...
0
votes
1answer
10 views
Showing that two random variables are standard normal but are not bivariate normal
Question:
If $X\sim \mathcal{N}(0,1)$ and we define $Y$ such that $$ Y = \begin{cases} X& \text{ if }|X|<a \\ -X& \text{ if }|X|\geq a. \end{cases} $$ Show that $Y\sim \mathcal{N}(0,1)$ ...
3
votes
0answers
32 views
Evaluate $I(a,0)$
Let $I(a,b)=\int_{0}^{1}x^a(1-x)^bdx$, where $a \geq 1, b \geq 1$ and $a, b$ are integers. If $b=0$ and $(1-x)^0=1$ (this is to confirm the property $x^0=1$) for x in $[0,1]$, then what is $I(a,0)$? I ...
2
votes
1answer
30 views
Criterion of map $M_f\to Z$ to be continuous
I was about to prove the following proposition
A map $M_f\to Z$ is continuous if and only if the induced maps $X\times I\to Z$ and $Y\to Z$ are both continuous.
Here, $M_f$ is a mapping cylinder of $...
