Sarcastic, Snarky and Skeptical

Is there a side to you that is sarcastic, snarky and skeptical? It doesn’t bother us, and we want you to make use of these gifts.

Math persons have skeptical attitudes and that’s why so often after seeing something that already looks reasonable, someone will say “Prove it!”

As you journey through Tensorworld, several times you will come to a junction where the textbook says “we can do ABC because XYZ is true.”

One of those composition notebooks from the grocery store and have a place where you keep a log of every “because ZYZ is true”.

Math is flexible. Think sbout all those wonderful truths we read in Euclid’s “Geometry” that are no longer true when we go to a place where things are nonlinear.

If someone says something is true, and you respond with, “well it’s only true for now…”, that’s snarky. Part of your Genius is realizing that we change the truth by changing the rules.

Skeptical shows up when you do tedious work to prove that something is true for the most general case. ” I won’t believe it is true until I see something that proves it is true” and I will then be asking myself “What could I do to change the truth?”

Sarcasm is defined as “harsh or bitter derision”. It goes beyond the lighthearted “Oh, by the way, you are wrong on this one.” Sarcasm is a growling animal that is ready to fight you.

Sarcastic is someone disagrees with you so much that every time you say something their first thought is: “okay, what could I change to make what was said untrue?”

Sarcastic is helpful. We want for you to ask yourself “what would it take to change the truth ?”

  • We want you to know how to make true into false (or vice versa)
  • We want you to understand the interface between true and false

Cautionary note: if something is true or false then anything that is interface has to be true or false.

Appendix A

\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} b_{11} + a_{12}b_{21} & a_{11} b_{12} + a_{12} b_{22} \\ a_{21} b_{11} + a_{22}b_{21} & a_{21} b_{12} + a_{22} b_{22} \end{bmatrix}

\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} b_{11} a_{11} + b_{12}a_{21} & b_{11} a_{12} + b_{12} a_{22} \\ b_{21} a_{11} + b_{22}a_{21} & b_{21} a_{12} + b_{22} a_{22} \end{bmatrix}

Let’s start with the northwest components:

a11b11 + a12b21 = b11a11 + b12a21

This is true if

a12b21=b12a21 [1]

Lets look at the southeast components:

a21b12 + a22b22 = b21a12 + b22a22

Because of [1l

a22b22 = b22a22 [2]

(Actually this would still be true even without [1])

Lets look at the southwest components:

a21b11 + a22b21 = b21a11 + b22a21

a21 occurs twice and b21 occurs twice. We can make the above equality true with b11=b22 [III] and a11=a12 [II]

The same thing happens in the northeast coordinate:

a11b12 + a12b22 = b11a12 + b12a22

Again: a11=a22 and b11=b22

Let’s go back to our first equality:

a12b21=b12a21

We dont see any a11,a22,b11,b22 so there is no way to use this with [II] or [III].

We can use [I] to calculate b factors as functions of other factors and we show this below before explaining why we did that:

\begin{bmatrix} a & a_{12} \\ a_{21} & a \end{bmatrix} \begin{bmatrix} b & \dfrac {a_{12}} {a_{21}} b_{21} \\ \dfrac {a_{21}} {a_{12}} b_{12} & b \end{bmatrix}

This is one way to make 2×2 commutative matrices. We have another:

for future content

We do notice something

  • a11b11 + a12b21 = b11a11 + b12a21
  • a11b12 + a12b22 = b11a12 + b12a22
  • a21b11 + a22b21 = b21a11 + b22a21
  • a21b12 + a22b22 = b21a12 + b22a22

Make the following true:

a12=0, a21=0, b12=0, b21=0

Any term with one if these will disappear:

  • a11b11 = b11a11
  • 0 = 0
  • 0 = 0
  • a22b22 = b22a22

The first and fourth equation are Tautologies (it is impossible for them to be false). Therefore, all values a11, b11,