User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)


 * $u \ v \ \mathsf{u} \ \mathsf{v} \ \nu \ \upsilon$

Anyone else have a hard time distinguishing between $u$ and $v$? I would like it to look more like this, does it confuse anyone else? It seems PW doesn't have the upgreek package. --GFauxPas 07:49, 27 January 2012 (EST)


 * Nope. Multiple years of extensive TeX writing and reading have trained my eye. I agree that referenced $v$ looks more distinguished, but imagine it is hard to implement. --Lord_Farin 08:08, 27 January 2012 (EST)

Exponential Definitions
I am discussing the equivalence of the definitions of exponential here:

http://forums.xkcd.com/viewtopic.php?f=17&t=80256

For anyone who has been following my progress or lack thereof on exponent combination laws/log laws etc, feel free to look on. --GFauxPas 16:59, 6 February 2012 (EST)


 * Okay, it looks like $e^{xy} = e^xe^y$ was the hardest one to prove! I was expecting a walk uphill the whole way. Oh, my Linear Algebra book came in the mail, so I guess I'll work on vectors next. And one of these days I'll have to tie up loose ends with Tarski. --GFauxPas 16:57, 10 February 2012 (EST)

Matrices
I was under the impression that column matrices and vectors were synonomous. But according to the definition on PW,


 * $\begin{bmatrix}

2 \\ 1 \\ e \end{bmatrix}$

is defined as:


 * $\left({1,1}\right)\mapsto2, \ \left({1,2}\right)\mapsto 1, \ \left({1,3}\right) \mapsto e$

But the vector $\left({2,1,e}\right)$ is defined as:


 * $1 \mapsto 2, \ 2 \mapsto 1, \ 3 \mapsto e$

Is it a difference that makes no difference?

Also, I have been thinking about matrices as, e.g.,


 * $\begin{bmatrix}

1 & 5 & \pi \\ 5 & 0 & \pi \\ 1 & 1 & -1\\ 6 & 100 & 2 \\ \end{bmatrix}$

As a vector $\left({\mathbf{u}, \mathbf{v}, \mathbf{w}, \mathbf{x}}\right)$ where:


 * $\mathbf{u} = \left({1,5,\pi}\right), \mathbf{v} =\left({5,0,\pi }\right), \mathbf{w} = \left({1,1,-1}\right), \mathbf{x} = \left({6,100,2}\right)$

Have I been wrong? Is it a difference that makes no difference? --GFauxPas 00:41, 12 February 2012 (EST)


 * My understanding is that:


 * $\begin{bmatrix}

1 & 5 & \pi \\ 5 & 0 & \pi \\ 1 & 1 & -1\\ 6 & 100 & 2 \\ \end{bmatrix}$


 * is not a vector. Not exactly sure "what" it "is" as such, apart from "er, it's a matrix, innit".


 * I also understand that:


 * $\begin{bmatrix}

2 \\ 1 \\ e \end{bmatrix}$


 * is a convenient technique for expressing the elements of a vector in matrix form. In this form it is straightforward to use the conventional mechanical technique for matrix multiplication, which is why it is taught in grade school (well, it was in mine). But as such it is only a notation for expressing the sizes of the elements of the vector - as it does not include the actual "unit vector" elements themselves (i.e. $\mathbf i, \mathbf j, \mathbf k$ or however you care to label them) it is not a "vector" as such.
 * You can also refer to the elements of the vector as $\left({2,1,e}\right)$ if you like, and many sources do as it's more typographically convenient.
 * As you say it's a difference that makes no difference. --prime mover 01:33, 12 February 2012 (EST)


 * It is convenient that matrices are well-behaved when thought of as 'rows of vectors' or 'vectors of rows'. If you don't care about the axioms too much (there is a quite canonical isomorphism anyway) these are the most useful ways to think about matrices. --Lord_Farin 19:01, 12 February 2012 (EST)