ALG
  • Member for 3 years, 3 months
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  • Milano, MI, Italia
Starting with a false statement, how can one prove anything is true?
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4 votes

just for fun! assume there exist $a,b$ relative prime integers such that $\frac{a}{b}=\sqrt{2}$, we can assume $a$ odd (otherwise we can argue in a similar way with $b$) hence $a \;\text{mod} \;2 =1$....

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quant interview: (mathematical modelling) linear regression and statistical significance
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4 votes

You can use the $F$-test in order to calculate statistically significance, given you hypotheis you have that $$ F_n= \frac{\rho^2}{1-\rho^2}*(n-2) $$ Hence you obtain the following $F_{100} = 0.0098$...

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Minimum number of vectors that span a linear subspace
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3 votes

the last equation tell you that also $\mathbf{v_4}$ is not necessary since $$ \mathbf{v_4} = -\frac{2 \mathbf{v_3} + 7 \mathbf{v_2} + 4 \mathbf{v_1}}{2} $$ hence it lies in $\mathcal{L} \{ \mathbf{...

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Matrices question - problem with task or with my understanding?
2 votes

your hypothesis is that the equation $AB=B$ holds for any matrix not only for the null matrix. Try to use different matrix such as for $B$ take the identity matrix or any invertible one

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Expectation of total scores when rolling a die until the score is not 6
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2 votes

The expected sum after $k$ rolls is not $k$ but $6(k-1)+3$ indeed if we stop at first roll it means that we get $1,2,3,4,5$ and the average sum will the mean i.e. $3$, if we rolled twice means that ...

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Conjugate elements in a group
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2 votes

In general it is false, just consider the cases in which $H$ is abelian when two different elements cannot be conjugates. For a more concrete example let us consider $S_3$ the group of permutation of ...

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How can I to prove the equality of intervals of open intervals with the equality of the closed interval
2 votes

Use double inclusion, clearly $[0,1]$ is contained in the intersetion since each point $P\in [0,1]$ lies in each open interval $(\frac{-1}n ; 1+\frac{1}n)$. For the other inclusion, pick a point $Q$ ...

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$A$ and $U$ are homeomorphic; $U$ is open in $X$ $\Rightarrow$ $A$ is open in $X$?
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2 votes

Without any other assumption the sentence seems to be false. Consider the topological space consisting of a line $l$ and one point $P$ external to the line. Then $P$ is open in the total space, ...

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Are there infinitely-many numbers that are both square and triangular?
2 votes

Have a look at https://en.wikipedia.org/wiki/Square_triangular_number So, the answere is yes!

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Prove that if $ab=k$ then if we times a by 2 we have to divide b by 2.
2 votes

Written in a such way the problem is quite trivial, clearly $$ ab = 2a\frac{b}{2}, $$ hence if $ab=k$ we get that $ 2a\frac{b}{2}=k$ (unless you are in char =2, but I think it is not the case). So, ...

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Does this result for inverses of linear transformation holds for infinite dimensional case.
1 votes

In general the statement is not true. The problem is that $S_1T$ is the identity on the image of $T$ which could be smaller than $V$. Think for example in the space of polynomial ring the ...

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Linear regression model when intercept is known
1 votes

For simple linear regression you the formula: \begin{align} \beta_1 &= \frac{\sum_{i=1}^N (x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^N (x_i-\bar{x})^2}\\ \beta_0 &= \bar{y}-\beta_1 \bar{x} \end{...

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Octal palindromes with even number digits are all composite numbers?
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1 votes

let us consider any number in base 8, $a_n 8^n +,\dots, + a_0$ observe that if $n$ is even then $ a^n \equiv 1 \;\text{mod} 9$ and if $n$ is odd then $ a^n \equiv -1\; \text{mod} 9$ then write the ...

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The family of all subsets of $X$ that contain a fixed set $Q$ is regular under what conditions?
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1 votes

If $Q= \emptyset$ the $\tau$ is the discrete topology i.e. each set is open hence $X$ is regular. Otherwise each pair of open set $A$ and $B$ clearly have not trivial interception ($Q\subset A\cap B$) ...

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Diagonalization of the matrix of a linear application and search for a base.
1 votes

By definition eigenvector satisfies the equation $$ A(v) = \lambda v $$ So you just have to find the kernel of the matricx $A-\lambda I$, where $\lambda$ is one eigennvalue and $I$ the identity ...

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How to find a particular number in a triangular arrangement of numbers
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1 votes

The first element of the $n$ th row will be $\frac{n(n-1)}{2}+1$, since it will be the sum of all diferences plus 1, then if you want the $k$ the element of the $n$ th row it will be $\frac{n(n-1)}{2}+...

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Use Induction to prove the following generalisation of the triangle inequality:
1 votes

You want to show that $|a_1+\dots+a_n+a_{n+1}|\leq |a_1|+\dots +|a_{n+1}|$. Using induction we have that $|a_1+\dots+a_n+|\leq |a_1|+\dots + |a_n|$, hence we get $$ |a_1+\dots+a_n+a_{n+1}|\leq |a_1 +\...

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What is the t-score for the relationship between X & Y
0 votes

The $t$-score for a parameter $\beta_i$ is defined as $t_i = \frac{\hat{\beta_i}}{s_{\hat\beta_i}}$, in your case you should find the $t$-score of the slpoe $t=\frac{0.15}{0.0005}$

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Relationship between order of a group acting on a set and faithfulness
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I $G$ acts faithfully on a set $X$ then we can define a new action on any set $X\subset Y$ setting $g(a)=a$ for any $a\in Y\setminus X$ and $Y$ can have any cardinality

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Find the Jordan canonical form
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0 votes

Since $\dim\text{Ker}(N^4)= 13$ and $\dim\text{Ker}(N^5)= 15$ the maximal Jordan block is of size 5, and you have exactly two Jordan blocks $J_1$ and $J_2$ of size 5, indeed, if there are no such ...

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What is the value of z in regards to the question?
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0 votes

If you want the 95% of the population in a gaussian you have two tails, so you need to consider the popultaion between 2.5% and 97.5%

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How do i start this indices question?
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0 votes

try to find what is missing from $B$ to be a cube, i.e. $p$ has exponent 1 so since it is a prime number you need to multply for $p^2$ so that the prodocut is $p^3$ that is a cube, in the same way you ...

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conclude that $x^{|G|} = 1_G$ by taking $H = \langle x\rangle \subset G$?
0 votes

Observe that $H$ is a cyclic gruop hence the order of $x$ is $n=|H|$ which implies that $x^n=1_G$ hence $x^{|G|}=1_G^{|G|/n}=1_G$

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Linear independence of finite dimentional vector space for a linear transform raised to a power
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If $m=1$ then there is nothing to prove, otherwise let us assume they are linearly dependent, then we have: $$\alpha_0 v + \dots + \alpha_{m-1} T^{m−1}v =0$$ Applying $T^{m-1}$ on both sides we ...

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Showing that $\hat \beta_1 = S_{xy}/S_{xx}$ for a simple linear regression
0 votes

In order to find $\beta_0$ and $\beta_1$ you have to solve the linear system $$\begin{align} \frac{\partial L}{\partial \beta_0} &= -2\sum_{i=1}^N (y_i-(\beta_0 + \beta_1 x_i))=0,\\ \frac{\...

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Matrix Representation of Linear Transformation from R2x2 to R3
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0 votes

Start calculating the image of the basis $A$, for example for the first element of the basis we have $$T\left(\begin{bmatrix}1&0\\0&0\end{bmatrix}\right) = (1,0,0)$$ then write the element as ...

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Matrix/vector proof
0 votes

If it helps, you can start with a particular matrix, however in order to prove the statement you need to make it in general finding excplicty the vector $x$. For the first point if you pick $x=0$ ...

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How can I prove $\hat\beta_0$ and $\hat\beta_1$ are linear in $\hat Y_i$?
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0 votes

If you are looking for the OLS solution as best fit for your model, then the formula for finding the vector $\beta$ is given by \begin{equation} \beta =(XX^T)^{-1} X^T Y \end{equation} provided ...

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