Detexify

Draw a symbol and get the corresponding code in latex,

http://detexify.kirelabs.org/classify.html

There exists a unique polynomial of degree $n$ passing through $(n+1)$ points in a plane.

So, if one we have a one-point say $(a,b)$ then clearly $f(x)=b$ is the (unique) polynomial of degree $0=1-1$ through the point. Now, we all know, $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$ is the unique polynomial (put $y=f(x))$ of degree 1.

Hence we sort of asking the general question and fortunately, the answer is yes, which one proves using the Vandermonde matrix and of course Lagrange Interpolation formula actually gives the polynomial.

Here is the solution,

Reference:
https://math.stackexchange.com/questions/837902/prove-there-exists-a-unique-n-th-degree-polynomial-that-passes-through-n1-p


Inverse of a Vandermonde Matrix : ​https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix 

Alguns Provas

https://www.dropbox.com/s/hgk6soslss5gvhz/P2-T5.pdf?dl=0

Gabarito Mat 137 e 140

https://www.dropbox.com/s/xci591aw40b7pgo/gabarito_exame_final_mat137.pdf?dl=0

https://www.dropbox.com/s/sfis345m7h9lojz/prova_gabarito%20%282%29.pdf?dl=0

https://mega.nz/#!dUNDQIKD!HVQwXJgNv2EwwIPypCu0A6s_v0NdFPlX8-wfAfpi_1Y

Plot a graph in $\LaTeX$

Here is the code that you might like to use to plot a graph in $\LaTeX$

$\begin{figure}[h!]

     \centering

     \begin{tikzpicture}[scale=0.70]

     \draw[thick, ->] (-12, 0) -- (12, 0);%asse x, la punta puo' essere > 

     o latex ( o altra roba ma..)

     \draw[thick, ->] (0, -12) -- (0, 12);%asse y

     \draw[thick, dotted] (1,-12) -- (1,12);

     \draw[thick, dotted] (2,-12) -- (2, 12.0);

     \draw[thick, dotted] (3,-12) -- (3, 12.0);

     \draw[thick, dotted] (4,-12) -- (4, 12.0);

     \draw[thick, dotted] (5,-12) -- (5, 12.0);

     \draw[thick, dotted] (6,-12) -- (6, 12.0);

     \draw[thick, dotted] (7,-12) -- (7, 12.0);

     \draw[thick, dotted] (8,-12) -- (8, 12.0);

     \draw[thick, dotted] (9,-12) -- (9, 12.0);

     \draw[thick, dotted] (10,-12) -- (10, 12.0);

     \draw[thick, dotted] (11,-12) -- (11, 12.0);

     \draw[thick, dotted] (-1,-12) -- (-1, 12.0);

     \draw[thick, dotted] (-2,-12) -- (-2 , 12.0);

     \draw[thick, dotted] (-3,-12) -- (-3, 12.0);

     \draw[thick, dotted] (-4,-12) -- (-4, 12.0);

     \draw[thick, dotted] (-5,-12) -- (-5, 12.0);

     \draw[thick, dotted] (-6,-12) -- (-6, 12.0);

     \draw[thick, dotted] (-7,-12) -- (-7, 12.0);

     \draw[thick, dotted] (-8,-12) -- (-8, 12.0);

     \draw[thick, dotted] (-9,-12) -- (-9, 12.0);

     \draw[thick, dotted] (-10,-12) -- ( -10.0,12);

     \draw[thick, dotted] (-11,-12) -- (-11, 12);

     \draw[thick, dotted] (12,1) -- (-12, 1);

     \draw[thick, dotted] (12,-1) -- (-12, -1);

     \draw[thick, dotted] (12,2) -- (-12, 2);

     \draw[thick, dotted] (12,-2) -- (-12, -2);

     \draw[thick, dotted] (12,3) -- (-12, 3);

     \draw[thick, dotted] (12,-3) -- (-12, -3);

     \draw[thick, dotted] (12,4) -- (-12, 4);

     \draw[thick, dotted] (12,-4) -- (-12, -4);

     \draw[thick, dotted] (12,-5) -- (-12, -5);

     \draw[thick, dotted] (12,-6) -- (-12, -6);

     \draw[thick, dotted] (12,-7) -- (-12, -7);

     \draw[thick, dotted] (12,-8) -- (-12, -8);

     \draw[thick, dotted] (12,-9) -- (-12, -9);

     \draw[thick, dotted] (12,-10) -- (-12, -10);

     \draw[thick, dotted] (12,-11) -- (-12, -11);

      \draw[thick, dotted] (12,5) -- (-12, 5);

     \draw[thick, dotted] (12,6) -- (-12, 6);

     \draw[thick, dotted] (12,7) -- (-12, 7);

     \draw[thick, dotted] (12,8) -- (-12, 8);

     \draw[thick, dotted] (12,9) -- (-12, 9);

     \draw[thick, dotted] (12,10) -- (-12, 10);

     \draw[thick, dotted] (12,11) -- (-12, 11);

     \draw (12,0) node[below]{$x$};

     \draw (0, 0) node[below left]{$0$};

     \draw (0,12) node[left]{$y$};

     \end{tikzpicture}

    \end{figure}$

Gmail - Attach email to email!

Ever dreamt of sending a set of emails to another email as an attachment. In case, you did here is the answer to your dream,

https://www.theverge.com/tldr/2019/12/9/21003638/google-gmail-email-attachement-feature-update 

Useful tips to visit Europe

Here is the blog which I found useful for the subject

https://www.nomadicmatt.com/travel-blogs/cheap-ways-to-travel-across-europe-2/ 

One more thing which might suggest the right option for travel within Europe.

https://www.omio.com/ 

Well, I personally like very much the following site for the bus tour,
https://www.flixbus.pt/?_sp=77e2d29c-1cbd-45cc-8b82-b330f1bb1c52.1576447316263&atb_pdid=479e6a8f-1711-4c02-b77d-a1563465a5ce&wt_eid=2157641830588589778&wt_t=1576447316332


Talking about visiting another country, the internet is something very crucial, and that leads to buying a sim card. Well, there are infinitely many agencies begging you to buy their insanely costly sim. I usually prefer buying one in the country I visit. Here is something you might like to do in case you agree with me and you are touring Europe,

https://www.eurocheapo.com/blog/best-sim-cards-for-your-trip-to-france.html

$AA'$ and $A' A$ have same set of eigenvalues.

Suppose $\lambda$ is an eigenvalue of $AA'$. Hence
$\begin{eqnarray*} AA' \cdot x &=& \lambda\cdot x\\A'A(A' \cdot x) &=& \lambda A'\cdot x\\ \end{eqnarray*}$
This proves, $\lambda$ is an eigenvalue of $A'A$.

A Different Way to Solve Quadratic Equations

No more buying EUROS while visiting Europe!

Friends,

One need not have to buy euros anymore while visiting Europe. There is a bank called N26. There are some countries where you can open this account (Standard account is for free) and they will deliver the card at your doorstep. In Brazil, the bank does not deliver the card currently but soon will be functioning here, as in the news. Although you recede in Brazil, you can open an account as far as you have some address in Europe. You just have to put residency in Europe while you fill in the open account form.

It is an easy process to open an account. Once you are done with it, just make a transfer using an international transfer service to your account. I myself like, TransferWise, in case you want to get yourself a first free transfer, use the following invitation link,

https://transferwise.com/invite/u/bhavinkumarm


Have a nice flight!

Fly cheap all over the world!

Hey friends!

Yeah, we always search for some way to not spend a lot of money while flying. I fly often and hence had and always will have this question of how to get a bang for the buck! Well, I often (PS. In incognito mode) search flights on some sites (written as per my preference),

1. Google Flights and Hopper (This app is just awesome! They have really good tech support.)

2. https://www.kiwi.com/us/

3. Kayak

Furthermore, there is often a problem packing bags while flying. Here is something that might help you,

Hand luggage: This packing trick taught in the Army could beat airline restrictions | Travel News | Travel | Express.co.uk

4 Easy Ways To Use Google Flights to Find Cheaper Flights And Organize Your Trips,

​https://www.forbes.com/sites/advisor/2019/12/13/4-easy-ways-to-use-google-flights-to-find-cheaper-flights-and-organize-your-trips/


5. Go for https://www.seatguru.com/ for reserving your place. It will give you all the information you will ever need to get the best possible seat. Do check-in as soon as it opens. Have a nice flight :)

6. In case you might like to track your flight, you might like to see the following link

https://thepointsguy.com/guide/how-to-track-airplane/

Linear algebra and Machine learning

Check out an awesome article here,

https://machinelearningmastery.com/examples-of-linear-algebra-in-machine-learning/

Take your machine learning skills to another level

https://mlwhiz.com/blog/2019/12/25/prod/

Linear Algebra in Data Science
https://www.analyticsvidhya.com/blog/2019/07/10-applications-linear-algebra-data-science/

Alguns Gabaritos - Matemática UFV

Podem encontrar alguns Gabaritos no link,

Gabaritos

Score a flight upgrade using two simple words

See the following article.

https://fortune.com/2018/02/22/flight-upgrade-tips/amp/

Two matrices with same characteristic polynomial need not be similar.

We all know that any two similar matrices have the same characteristic polynomial. Naturally, we might ask for the converse. Unfortunately, the converse is not true and a counterexample on can find at the following link,

http://www.mathcounterexamples.net/two-non-similar-matrices-having-same-minimal-and-characteristic-polynomials/

Top of the place for the answer, above site, in my opinion, seems to be a great site to find counterexamples in various disciplines.

Well, as I thought, some of you might be thinking that $A$ and $A^T$ might serve as a counterexample. In fact, $A$ and $A^T$ are similar always. One can see this on,

https://math.stackexchange.com/questions/62497/matrix-is-conjugate-to-its-own-transpose

As explained on the link, if $A$ happens to be diagonalizable it is extremely easy to conclude.

Every matrix over $\mathbb{C}$ is diagonalizable?

The answer is NO and can be found at the link,

https://math.stackexchange.com/questions/1926798/are-matrices-ever-diagonalizable-in-complex-field

In case the link doesn't work, see

Simple tips to avoid a worst nightmare of luggage loss while flying

• Check out the link below,

https://www.express.co.uk/travel/articles/1211172/Luggage-lost-simple-mistake-on-flights-travel-expert-insurance

$AB$ and $BA$ have same set of eigenvalues.

If $0$ is an eigenvalue of $AB$ then clearly, $\det(AB)=0$, which as a result (without loss of generality) gives $\det(A)=0$. Hence $\det(BA)=0$, as a result $0$ is an eigenvalue of $BA$.

Let $\lambda\neq 0$ be an eigenvalue of $AB$ with $x\neq 0$ and $ABx=\lambda \cdot x$. Let $Bx=y$. If $y=0$ then $\lambda \cdot x=ABx=Ay=0$. Hence $\lambda=0$ or $x=0$, a contradiction. So we have  $y\neq 0$.

$\begin{eqnarray*} BAy &=& BABx\\&=& B\lambda \cdot x \\&=& \lambda Bx\\&=& \lambda BAy. \end{eqnarray*}$

That proves $\lambda$ is an eigenvalue of $BA$ with $y=Bx$ a corresponding eigenvector.

One can prove the same fact in another way as follows, if (without loss of generality) $A$ is invertible.

Note that, $AB = ABAA^{-1}$, which proves $AB$ and $BA$ are similar matrices and any two similar matrices have the same eigenvalues. 

Brazilian travellers will soon be able to use global entry program

https://thepointsguy-com.cdn.ampproject.org/v/s/thepointsguy.com/news/brazilian-travelers-will-soon-be-able-to-use-global-entry/amp/?amp_js_v=a2&amp_gsa=1&usqp=mq331AQCKAE%3D#referrer=https%3A%2F%2Fwww.google.com&amp_tf=From%20%251%24s&ampshare=https%3A%2F%2Fthepointsguy.com%2Fnews%2Fbrazilian-travelers-will-soon-be-able-to-use-global-entry%2F

Latex Templates for various types of writings

Friends,

The latex is the best thing to make write up of anything. I find it absolutely stunning for the purpose. Here are some templates that you might like to use as per your requirement.

https://www.latextemplates.com/

The template I use to make exam papers is,

https://mega.nz/#!IMdXCIbZ!VMoa9x2ag1uT14itPewLVq2BRE4u7naJfUovHwTWTN0

Happy LATEXING!

Gilbert Strang: Linear Algebra, Deep Learning, Teaching, and MIT OpenCou...

"Linear algebra is flat while calculus is curved - Prof. Gilbert Strang". I guess one must watch this awesome video.

https://youtu.be/lEZPfmGCEk0

Indian passport holder no more need transit visa while passing through Paris.

Friend,
Looks like Indians no need Transit Visa. See the links below,

https://www.schengenvisainfo.com/news/indians-no-longer-require-airport-transit-visa-in-france/

https://www.schengenvisainfo.com/france-visa/airport-transit-visa/

However, there is a disclaimer there

https://www.schengenvisainfo.com/terms-conditions/

So get enough information before heading for it. 

Sum and product of eigenvalues and eigen values of the matrix transpose

The sum (product) of the eigenvalues of a matrix is a trace (determinant) of a matrix. One can find the proof at the following link on page 3,

https://www.adelaide.edu.au/mathslearning/play/seminars/evalue-magic-tricks-handout.pdf

The matrice $A$ and $A^{T}$ have the same set of eigenvalues, a proof can be found at the same link on page 2.

PS. The sum and product are done taking multiplicity into account, as can be seen in a proof provided by the link.

My Movie list

Charging phone via USB in public place can be dangerous!

The following link might be of interest to you,

https://www.melhoresdestinos.com.br/carregar-celular-aeroporto-juice-jacking.html?utm_source=Promo%C3%A7%C3%B5es+de+Passagens+A%C3%A9reas&utm_campaign=bca3b488b5-news_diaria_md_COPY_01&utm_medium=email&utm_term=0_4a12f3849e-bca3b488b5-63085205&mc_cid=bca3b488b5&mc_eid=17cc200473

$\int \frac{5 x^{3}+12}{x^{3}-5 x^{2}+4 x} d x$

Note that, $5 x^{3}+12 = 5* (x^{3}-5 x^{2}+4 x)+ 25x^2-20x-12$

$$\begin{eqnarray*} \int \frac{5 x^{3}+12}{x^{3}-5 x^{2}+4 x} d x &=&\int 5 \ dx+\int \frac{25x^2-20x-12}{x^{3}-5 x^{2}+4 x}\\ &=& 5x + \int \frac{A}{x}+ \int \frac{B}{x-4}+ \int \frac{C}{x-1}\\ &=& 5x -3\ln|x| + \ \frac{77}{3}\ln|x-4| \ + \frac{7}{3}\ln|x-1| +\ C \end{eqnarray*}$$

Ex 24 da lista

24. a) Teorema de Núcleo e Imagem (TNI) $\Rightarrow \ Dim(Im T)=3$. Então $Det\left(\begin{pmatrix}1&2&-1\\a&0&1\\1&b&c\end{pmatrix}\right)=-b-2ac+2-ab\neq 0$.

b) TNI $\Rightarrow \ Dim(Im T)=2$. Então $Det\left(\begin{pmatrix}1&2&-1\\a&0&1\\1&b&c\end{pmatrix}\right)=0$

c) TNI $\Rightarrow \ Dim(Im T)=1$. Note que, $\{u_1, u_2\}$ é independente para todos $a$. Então, $Dim(Im T)\ge 2$. Então, não existe os valores de $a,b,c$. 

Invertibiliity of a square matrix and characteristic polynomial

Here is a nice article on a subject,

https://math.stackexchange.com/questions/166219/is-a-matrix-with-characteristic-polynomial-t2-1-invertible

In fact, as shown in the following photo, one can prove the statement next to the highlighted statement easily.

A proof goes as follows,

It is clear that $\lambda$ is an eigenvalue of $A$ iff $Det(A-\lambda I)=0$. As a consequence, $\lambda=0$ is an eigenvalue of $A$ iff $Det(A-0\cdot I)=|A|=0$. Hence, $|A|\neq 0$ iff $\lambda=0$ is not an eigenvalue of $A$ iff the characteristic polynomial of $A$ does not have zero at zero.

Fermat's Last Theorem - The Theorem and Its Proof: An Exploration of Iss...

A very captivating, inspiring and very stunning documentary that everyone must watch again and again,





One must see the following video too, which is on Abel Prize Winner Prof. Andrew Wiles.

https://www.youtube.com/watch?v=cWKAzX5U85Q

1. What is Computation? - PYTHON

A nice course on Python for beginners,

Lecture 2: Story Proofs, Axioms of Probability | Statistics 110

A nice course on Probability

Massive Open Online Courses (MOOCS)

I will be posting education, mostly mathematics and computer science related stuff on this page. Feel free to comment and email me.

Of course, there is a long list of sites that offer a free course today. Following is one of the best site that I found so far offering the free courses and a lot more. Let me know if you have some other suggestions on this.

http://www.openculture.com/

https://ocw.mit.edu/courses/find-by-topic/

As on 11th March, Harvard University is offering a free course on Mathematics and Data Analysis.

https://online-learning.harvard.edu/

Reflection of a point $(x,y)$ in a given line $ax+by+c=0$

Friends,

The answer to the title of the post can be found here at the following link,

https://drive.google.com/file/d/0By83v5TWkGjvb2tuekNSUFo3cFE/view

Also, one can see alternative approaches to the same problem in the following link,

https://math.stackexchange.com/questions/1013230/how-to-find-coordinates-of-reflected-point

The very idea that one implements in the first link will be absolutely clear if one understands the following procedure,

Travel - Hotel Booking Discounts

HI,

I travel a lot. Here are some referral links which might get you some discounts on bookings,

1. Booking.com

No more referral link on Booking.

2. Airbnb.com

https://www.airbnb.com.br/c/bmoriya?currency=BRL

Use Latex in Blog

Hi,

In order to use latex in Blog, one needs to follow a few steps. I saw this nice video while solving the problem.

https://www.youtube.com/watch?v=sqz78Jnrgcs

https://www.codecogs.com/latex/eqneditor.php

Following are a few examples of the code which shows up right on the blog,


1. $$\begin{eqnarray} I &= & 1/3\int \frac{du}{\sqrt{u}(u+1)}\\ &= & 2/3\int \frac{2v\ dv}{v^2+1} \mbox { (Substituting } u=v^2 \mbox{we get } du = 2v\ dv)\\ &= & (2/3) \tan^{-1}(\sqrt{x^3-1}) \end{eqnarray}$$

2. $$\begin{equation} x=y\\ z=w \end{equation}$$

Calculus - MAT 140 (UFV)

In this post, we shall be posting some problems solved,

1.  $\int \frac{1}{x\sqrt{x^3-1}}dx$

Note that, $I = \int \frac{x^2}{x^3\sqrt{x^3-1}}dx$

Set $u=x^3-1$.

Hence $du=3x^2 \ dx$.

As a result we have,

$$\begin{eqnarray} I&=&1/3\int\frac{du}{\sqrt{u}(u+1)}\\ &=& 2/3\int \frac{v\ dv}{v(v^2+1)} \mbox { (Substituting } u=v^2 \mbox{we get } du = 2v\ dv)\\ &=&(2/3)\tan^{-1}(\sqrt{x^3-1}) \end{eqnarray}$$

​

Brazilian Money Transfer Apps

Friends,

I have been using the following money-transfer apps for the last few months. So far my experience is pretty good. I have referral codes for them, which will get you a discount on your registration.

1. PicPay - Use o PicPay para pagar amigos, boletos, recarregar o celular, Steam e muito mais. Ao criar sua conta, insira o meu código, QJBD37, e ganhe de volta os primeiros R$10 que você gastar usando seu cartão de crédito. Baixe agora! http://www.picpay.com/convite?@QJBD37

2. Recarga Pay - Oie! Estou te dando R$10 pra pagar qualquer boleto, recarregar o celular ou Bilhete Único no RecargaPay, o app número 1 do Brasil em pagamentos! É só usar o código bhamor170 ou o link http://recargapay.com.br/r/bhamor170-Zn

3. Mercado Pago - ​Você ganhou R$ 10 para o seu primeiro pagamento com cartão de crédito nos apps do Mercado Livre ou Mercado Pago.

Use este valor para pagar com código QR no McDonalds ou Drogasil, recarregar seu celular ou seu vale-transporte.

https://invite.mercadopago.com.br/mgm?booster_id=203470229

4. Banqi - 1. Just create your account using the link, https://app.banqi.com.br/Vs4tPVUUr1 

OR

                 2. You may as well use the code, 9NKR-VF8Q

5. ITI - com o iti vc paga pessoas e estabelecimentos de um jeito fácil. saia sem carteira, sem cartão, só com o celular na mão ;) acesse https://iti.itau?querobaixar

TransferWise International Money Transfer

Hi,

Friends living abroad always wonder about finding out some way which costs less money to send out money home. TransferWise (TW) is an exact solution to the problem. I have been using the service for the last four years and I must say I am extremely satisfied with the service they offer. All the doubts you might have, they have a very concise answer. Get yourself a discount on first transfer by registering using the following link,

https://transferwise.com/u/bhavinkumarm


Just Go for it. Happy TransferWise!!!

Some useful UBUNTU commands

Hi so in this post I am going to mention some useful commands,

1. To remove *.swp file in all subdirectories

Run the following command

shopt -s globstar
rm -f /path/**/*.swp

\\\ ** matches any file or folder in an arbitrary sub-directory. Note that * just matches files and folders in the root directory.

**/.* matches any file beginning with a '.' (usually hidden files, like the .git folder) in an arbitrary sub-directory.


**/*.@(jpg|jpeg|gif|png) matches any file in an arbitrary sub-directory that ends with a '.' and exactly one of either 'jpg', 'jpeg', 'gif', or 'png'.\\\



For example,
In order to remove say, "*.jpg" files in all subdirectories of the folder called "/home/UB", one needs to access the UB folder. Just run the following commands one by one,

ls UB/

shopt -s globstar

rm **/*.jpg

2. Compress the size of PDF file

The objective is the reduce the size of mypdf.pdf

Use cd to get to the directory and then run the following command.

Method 1. Run the following commands

pdf2pdf mypdf.pdf

ps2pdf mypdf.ps

du -hs mypdf.pdf %gives you the reduced size of mypdf.pdf

Method 1. Using GhostScript.

Install GhostScript by using the following command

sudo apt-get install ghostscript

Then run,

gs -sDEVICE=pdfwrite -dCompatibilityLevel=1.4 -dPDFSETTINGS=/screen -dNOPAUSE -dQUIET -dBATCH -sOutputFile=1small.pdf mypdf.pdf

%1small.pdf is a reduced version of mypdf.pdf

3. Play with pdf files

There are some wonderful things you can do with the pdftk and those can be found at the following link,

https://www.pdflabs.com/docs/pdftk-cli-examples/




PS. Things written after % are comments, i.e do not consider them as a command.