I never want to write another proof again [#68]

I just finished my homework for my linear algebra class. I wish both sides of my professor's pillow are warm tonight. I had to write ten pages of LaTeX for this homework. The homework is due weekly!! The worst part: it's all proofs!!!

This homework is insane. I'm taking four other classes this semester, and the weekly homework for my linear algebra class takes as long as the weekly work of my other four classes combined. That should not be happening! But let me explain why it's happening.

1

For those who don't know, proofs are by far the worst kind of mathematical exercise. You must construct an airtight logical argument for a given statement based on known properties and theorems, which is often infuriatingly hard. Here's one of my proofs:

Did you read it? The correct answer is no, because why would you do that to yourself? Hopefully you're seeing what I've gotten myself into, by virtue of being a math major.

My professor assigns 13-14 exercises from the textbook per homework. Almost every single one is a proof. How do I approach each one?

2

Here's a step-by-step process of how I approach each proof I write:

1. Copy-paste a couple of lines into my LaTeX document:

\section{X.X.XX} % exercise number from textbook
    \setcounter{equation}{0}
    \begin{proof}
    \end{proof}

2. Take a look at the given statement/theorem, which causes me to react in one of three ways:
   1. "my dog, who is so dumb that he regularly eats leaves because he likes the crunch, could believe this statement without a proof because it is so mind-bogglingly obvious"
   2. "HUH????????? HOW on earth do I prove this???"
   3. "ok this actually seems like a reasonable thing to prove"
3. Stare at the nearest wall for around 15 seconds to see if my intuition can come up with anything off the bat.
4. Realize I need to actually sketch out a proof in my notebook.
5. Start sketching some algebraic manipulations in my notebook for a couple of minutes.
6. Get stuck on part of the proof.
7. Look back through the textbook, desperate to find any theorems or properties I can use. Become extremely frustrated.
8. Consider using a forbidden proof technique!

9. Consider checking Instagram and Reddit. Then realize that, unfortunately, procrastination is futile, and that ignoring the proof will not make it disappear from my homework.
10. Get back to rereading the textbook, again, for any properties or theorems I may have missed.
11. Have a spark of inspiration that allows me to complete the proof. Feel like an absolute genius and briefly consider myself to be the best mathematician that has ever lived.
12. Finish sketching out the proof in my notebook.
13. Type the whole proof into LaTeX. This can either be short and easy or extremely long and tedious. Here's the portion of my .tex file for the proof I displayed earlier:

\section{1.1.25}
    We want to prove Theorem 1.4: Let $\mathbf{x}$ be a vector in $\mathbb{R}^n$, and let $c$ be a scalar. If $c\vec{\mathbf{x}} = \vec{\mathbf{0}}$, then either $c = 0$, or $\vec{\mathbf{x}} = \vec{\mathbf{0}}$.
    \setcounter{equation}{0}
    \begin{proof}
        We proceed by contradiction. Assume that there exists both:
        \begin{itemize}
            \item a scalar $c$ such that $c \neq 0$
            \item a vector $\vec{\mathbf{x}} \in \mathbb{R}^n$ such that $\vec{\mathbf{x}} \neq \vec{\mathbf{0}}$.
        \end{itemize}
        such that $c\vec{\mathbf{x}} = \vec{\mathbf{0}}$. We can expand $c\vec{\mathbf{x}}$:
        \[
            c\vec{\mathbf{x}} = c[x_1, x_2, \cdots, x_n]
        \]
        Because $\vec{\mathbf{x}} \neq \vec{\mathbf{0}}$, we know that \textit{at least one} entry of $\vec{\mathbf{x}}$ is nonzero. We call this nonzero entry $x_k$, where $1 \leq k \leq n$. Therefore:
        \begin{align}
            c\vec{\mathbf{x}} &= c[x_1, x_2, \cdots, x_k, \cdots, x_n] \\
            &= [cx_1, cx_2, \cdots, cx_k, \cdots, cx_k]
        \end{align}
        We know that the entry $cx_k$ must be nonzero, because $c$ and $x_k$ are both nonzero real numbers. (This is a property of real numbers: the product of two real numbers can only be 0 if at least one of the two numbers is 0.) Thus, $c\vec{\mathbf{x}}$ \textit{cannot} be equivalent to the zero vector, because two vectors are only equivalent if their corresponding entries are equal. Therefore, we have shown a contradiction, and proven the theorem.
    \end{proof}

14. Reread my proof one more time to make sure I did it correctly. I always end up skimming it extremely quickly and being like "yeah, looks good."
15. Rejoice!! For I have now completed ~7% of my homework!! Then promptly become very sad because I have another proof to do.

Whenever I complete the homework, I feel three extremely strong, simultaneous emotions:

• "LET'S GO!!!! I'm so cool! I'm the best! I'm done! I'm free!"
• "I need to sleep for twelve hours. I just completed a Herculean effort. Actually it was even harder than a Herculean effort because I bet Hercules never had to write proofs."
• "What is the most effective prank that I can pull on my professor without getting in trouble with the university?"

3

So that was a long rant. I hope it is evident by now that I do not like writing proofs. But there are definitely a few benefits to this class and its accompanying homework:

• I've gotten significantly better at using Vim. Significantly better. I consider myself a relatively new Vim user, but I've already picked up so much muscle memory that makes typing in any coding language (including LaTeX) far, far easier.
• It's probably good if I'm academically challenged in college. I will certainly encounter harder classes as I move through my degree curriculum, and I shouldn't just have easy classes all the time.
• It builds character:

Thanks for reading!