Tagged Questions
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 ...
1
vote
1answer
9 views
Connected $T_3$ space
I wanted to prove that a connected $T_3-$space containing atleast two points must be uncountable.
My attempt is as follows:
Let $X$ be a connected $T_3-$ space and let $x,y\in X$. Then there exist ...
0
votes
0answers
25 views
Sum of four numbers divisible by 4
So I know that if 2 divides (x+y), then the sum of the two integers, x and y are even and thus have the same parity. In the case of 4 divides (a+b+c+d), what am I looking for when examining the sum of ...
1
vote
1answer
21 views
Find a basis for $p \in P_2(\mathbb R) $ with $p(7) = 0$?
Find a basis for the subset: $$ S = \{\;p \in P_2(\mathbb R)\;\; |\;\; p(7) = 0\; \} $$
I'm not sure how to approach this question.
$$
p(7) = a_0 + 7a_1 + 49a_2 = 0
$$
$$
a_0 = -7a_1 - 49a_2
$$
$$
...
2
votes
2answers
37 views
If 12x ≡ 1 mod 85, find x?
I found x = 78 by brute force but I want to use some very basic Elementary Number Theory proofs to find the answer to this. I am stumped, please help me. I'm going crazy.
2
votes
2answers
35 views
Proving that the closed unit square in the plane is compact.
My thoughts on proving this statement is as follows:
Suppose $G_a$ is an open cover of $Q= [0,1] \times [0,1]$. For each $x$ in $[0,1]$, there is some ball around $x$ with radius $r_x$ such that it ...
2
votes
1answer
18 views
Probability density function of a summation of continuous random variables
Let $Z_{i} = \tau + X_{i}$, where $X_{i}$ is a exponential random variable ($X_i \sim \varepsilon(\lambda)$), $0<\tau, \lambda < \infty$
Assume $X_i$ are independent random variables.
Suppose ...
0
votes
0answers
17 views
Show that a matrix A must be invertible for the set of A*v in all real numbers to be linearly independent
A = [A$_{ij}$]$_{nn}$
Assume that the set S = {Av$_1$, Av$_2$, . . . Av$_n$} is linearly independent.
I have to show that A must be invertible.
What I've done is:
0 = c$_1$A$v_1$ + c$_2$A$v_2$ + . ...
1
vote
2answers
42 views
Prove that $\lim_{n \to \infty} \int_0^2 e^{ x^2 / n}\,{\rm d}x$ exists and evaluate it.
I need to show that this limit exists and then evaluate it. It is from a section on uniform convergence of sequences. I know that if $f_n \rightarrow f$ uniformly and each $f_n$ is integrable, then I ...
0
votes
0answers
6 views
System of Linear Equations (3x5) expressing answer in vector form.
Solve the system, and write answer in vector form.
a + 2b + c + 3d =10
-a + b + 3c - 4d = -15
8a + 7b - 4c + 36d = 131
I can't figure out which are the free variables.
1
vote
1answer
30 views
Quotient ring of a matrix
Let $F$ be a field. Consider the set
$$R = \left\{ \begin{bmatrix}
a & b \\
0 & c
\end{bmatrix} : a, b, c \in F \right\}$$
Think of a non-trivial two-sided ideal of $R$ and describe in ...
2
votes
2answers
24 views
Proving the continuity of a difference of functions.
Prove that if $f$ and $g$ are continuous at $x=a$, then $(f-g)$ is continuous at $x=a.$
I have $|f(x)-f(a)-g(x)+g(a)| = |(f(x)-g(x))-(f(a)-g(a))|$ so far. I wanted to use the triangle inequality on ...
0
votes
1answer
21 views
Finding conditions so that the solution of a second order ODE tends to 0
Let $$y(x)=Ae^{m_1x}+Be^{m_2x}$$ be the general solution to a second order linear homogeneous ODE, with $m_1>0$ and $m_2<0$. If $y(0)=y_0$ and $y'(0)=y'_0$ then find the relationship between ...
0
votes
1answer
36 views
How can I plot the sum of sin(na) on Wolfram Mathematica? [on hold]
I need to plot the sum of sin(na) on Mathematica. How do I do it?
3
votes
1answer
35 views
Determine all solutions of the congruence relation $2x^{121} + 22x^{36} + 21x^{30} + 2 \equiv 0 \pmod {77}$
My solution so far is as follows:
Using the Chinese Remainder Theorem, we can simplify the congruence to the simultaneous congruence of:
$$
\mathrm{Simultaneously: }\ \
\begin{cases}
...
0
votes
1answer
9 views
Calculate the confidence interval for a single variance given two variances (using pooled variance)
I am having trouble trying to interpret why I'm given some value for a homework exercise. It goes like this:
Suppose we're studying the time it takes for a certain industrial process to complete. ...
