User:Smforrestal


 * 1) Well the first thing that I learned about wikipedia is that you have to have an account to edit text. I once thought that one could just double click on an article and edit it. It may have used to be like that, but now you need an account before you can go through and change things.
 * 2) The second thing is wikiversity. I didn't even know that wikiversity existed. Another thing about it that I found that I did not know was the fact that you could put quizzes and stuff on the website. The quizzes can be graded and do not count against you. I found that pretty interesting.
 * 3) The third thing is that you can use "special characters" to write mathematical formulas, or other different types of characters.
 * 4) The fourth thing that I learned is that wikiversity can double as a classroom type thing. It has videos and links and stuff to help you.
 * 5) The final thing I learned is that you have to use math syntaxes when trying to write a formula or equation.

Horners scheme:


 * $$p(x) = \sum_{i=0}^n a_i x^i = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n,$$

Improved version of Wikipedia's definition of a cobweb plot. The page dedicated to the cobweb plot has very useful visuals, however, there are only links to the equations of what the graphs represent. I would like to five an equation in which one can relate an illustration of the graph.

The cobweb plot is a a fairly simple concept. Using a cobweb plot, it is possible to infer the long term status of an initial condition under repeated application of a map. If the cobweb point has an inward spiral, this represents a stable fixed point. While an outward spiral represents an unstable fixed point.

Once you have a function, lets say:
 * $$f(x) = -(x - 3)^2 + 5$$

and the line
 * $$ y = x $$

and graph them, you get an upside down parabola with a diagonal line going through it.


 * 1)  pick a statring point and trace it vertically to the function curve.
 * 2) from this point, move horizontally across until you reach the diagonal.
 * 3)  upon reaching this point on the diagonal, you trace this point vertically back to the function curve
 * 4) repeat the process as many times as you can until you can no longer do so on paper or the line becomes a solid point. Where it seems that you're lines can no longer be moved, you have found the fixed point.

Final Project
For my final project, I am choosing matrix norms. The wikipedia page on matrix norms is pretty good, but I feel like they could show some proofs as to how they got to where they were. I proposed to add a proof for the L1 norm and a simple small example on how to find the l1 and l∞ norms. I would also want to throw in a couple of examples on how to calculate the L1 L2 and L∞ norms. My instructor suggested that I show the proof of the L1 norm in hopes to better explain it to myself. I did all of my editing on my own page first and then I copied it onto the matrix norm page. Once on the matrix norm page, I quickly inserted a 4x4 matrix and calculated the max of it. After posting my proof, I have not seen any changes to it from an outside source yet. The only problems I ran into were using the book I had as a reference and not plagiarizing it. Proofs can be very difficult to redo, so I simply changed some variables around and try to keep the same flow of the main page. Initially i was going to prove the L∞ norm as well, however I ran into difficulty with it, so I just stuck to the original plan. My changes are documented below and also on the matrix norm page and I referenced the book I used for the proof. It may not seem like a lot of work because the proof is small, but this actually took about 5 hours to completely finish and polish up. I personally feel that even though I may not understand the proof that well, I deserve a good grade for attempting something that was very intimidating and I feel I did pretty well.

My proposal is to prove the l1 norm on the page.


 * $$ \left \| A \right \| _1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} |, $$ Eq1

is the definition of the l1 norm and my goal is to show why and how. The L1 norm is very easy to calculate in most cases. It is the max column sum of the matrix.

If I am able to find a understandable proof for the L∞ norm, I will include that to the page as well.

We want to prove the


 * $$ \left \| A \right \|_1 $$

Let us begin by first declaring a constant k if we let

k = :$$ \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} |, $$

then we would have:


 * $$ \left\|Ax \right \|_1 \le \left\|Kx \right\|_i, $$

thus
 * $$ \left\|A \right \|_1 \le K $$

We have showed that :$$ \left\|A \right \|_1 \le K $$, however, we can be more precise.

By writing this as an inequality we can demonstrate x as
 * $$ K = \frac{\left \| A x\right \| _1}{\left \| x\right \| _1}. $$

Now we will declare C as the column index for the maximum of Eq1. Let :$$ x = e^c $$ the Cth unit vector then :$$ \left\|X \right\| = 1 $$ and
 * $$ \left\|A \right\|_1 = \sum_{i=1} ^n | \sum_{j=1} ^n | a_{ij} x_{j} ||| $$ = :$$ \sum_{i=1} ^n | a_{ik} | = K $$

This proves that for the vector norm :$$ \left\|. \right\| $$ the operator norm is
 * $$ \left\|A \right\|_1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} |, $$