\documentclass{letter}

\signature{Dane}
\address{769 Shady Drive East \\ Apartment 303 \\ Pittsburgh, PA 15228}

\begin{document}

\begin{letter}{Brian Sabo \\ 6 Windsor Way\\ Coraopolis, PA 15108}

\opening{Hey Brian,}

I was so pleased to receive your letter. To be honest with you, I don't think I
have ever received a hand-typed letter before in the mail. It was a nice break
from the usual impersonal text messages the modern day is used to. 

I actually have a great story for you about a challenging class I took, and
how it frankly changed my career path.

I was in my junior year of my mechanical engineering degree when a graduate
class called 'Linear Algebra for Machine Learning' was announced it would be
offered. It was taught by a new assistant professor in the department named Ben
Erichson. He was more of a statistics background, so I was very curious how the
class would be taught. Plus, who isn't interested in machine learning?

Let me tell you about an early hiccup in that class. Did you know that the least
squares estimate, a.k.a. a linear fit, can be written like this?

\begin{equation}
    \min_{\mathbf{x}} (\mathbf{A}\mathbf{x} - \mathbf{b})^T(\mathbf{A}\mathbf{x}
  - \mathbf{b})
\end{equation}

where $\mathbf{x}$ is the set of parameters, $\mathbf{A}$ is the input data, and
$\mathbf{b}$ is the measured output. Basically, you choose $\mathbf{x}$ such that you
minimize the error when you multiply it by the input and subtract the expected
output. You can get the best linear estimator by doing:

\begin{equation}
  \hat{\mathbf{x}} = \left( \mathbf{A}^T \mathbf{A}\right)^{-1} \mathbf{A}^T
  \mathbf{b}
\end{equation}

If you do it for a two dimensional case, you'll find you get exactly the result
a linear fit in Excel will give you.

Anyways, the derivation of this and the proofs were all in the first 60 minutes
of class. My cocky ass didn't even know you could write vectors as bold small
letters at that point. This class became the most difficult class I had taken to
that point. But, in the struggle that class presented, the learning of difficult
concepts provided me a sense of accomplishment and satisfaction that no other
class previously had. I knew at that point that I needed to find somewhere I
could keep learning and tackle hard problems. For me, that was a PhD. I don't
use the material from that class on a day to day basis, but I still think about
it a lot.

So let me answer your question. When it comes to focusing on a concentration,
taking more harder classes or taking the easy road, I say take what classes you
find \textit{interesting}. Even if those classes are "difficult", they will be
easy because you won't mind working on them. To that end, I don't think
certificates or concentrations are too important. I'd ask your dad if he cares
about a concentration when hiring someone, or would rather have someone who's
curious, versatile, and challenges themselves. 

There is another important angle to consider when choosing classes. Certificates
and concentrations are attractive. But, consider that if you're choosing between
a class that'll give you a subtitle on your degree and one that you're curious
about, remember that this is the \textit{only} time you will get to choose. When
you finish school, you won't have an opportunity to take that 'interesting'
class again, or really any class. Cherish the fact that you're able to pursue
different courses now.

Workload be damned. You're a very smart guy---you'll figure it out. If you want
to do great things, they won't come easy, but they won't come at all if you
don't push yourself. It's a privilege to be able to do this difficult work.

\closing{Your cousin and friend,}

P.S. Proofread your prose. You have typos in there! Most people will only know
you through your writing, and you don't want to give a bad impression.

P.P.S. I love the Leo picture and am keeping it in the office :)

\end{letter}

\end{document}
