Mathematics

[s243a 0]

There doesn?t seem to be any area for mathematics so for now I will put everything in one thread. Yesterday I was thinking about generalizing matrix multiplication to higher dimensions. I know this has already been done before (A.K.A.) tensors but I don?t know anything about tensors. Anyway I realized that there are three common types of multiplication that can be done on matrices. They are element wise multiplication, the inner product and the outer product. Many types of matrix multiplications can be expressed as a combination of these. For instance standard 2D matrix multiplication is a combination of the inner and outer product. The rows of matrix one and the columns of matrix two are used in the outer product while the columns of matrix one and the rows of matrix two are used in the inner product.

[Zenchi 0]

o_o;.. s243a, can you give us people who are only taking Geometry a little bit of information on this? I'm sure I could help, if only I knew what you meant exactly.. I mean, I can grasp it, but then it floats away.There's no place like home.. there's no place like home..

[googlue 0]

o_o;.. s243a, can you give us people who are only taking Geometry a little bit of information on this? I'm sure I could help, if only I knew what you meant exactly.. I mean, I can grasp it, but then it floats away.

 

There's no place like home.. there's no place like home..

<{POST_SNAPBACK}>

Hey Birthday boy,

Forget geometry today and enjoy your day

[s243a 0]

Okay, well lets start with the basics. The inner product also known of as the dot product is the sum of element wise multiplication. For instance if you have two vectors [a b] and [c d] then the inner product is ac+bd and is denoted:

[a b] dot [c d]=ac+bd

The operation is done by first doing element wise multiplication which I will denote as .* for example:

[a b].*[c d]=[(a*B)  (c*d)]

And then by taking the sum of the resulting vector.

Sum_{over all elements}=(a*B)+(c*d)

The inner product is a measure of how close two vectors are, some applications include geometry and signal processing.

Matrix multiplication can be thought of as row by column multiplication. For instance.

Let:

A=[a11     a12][br]  [ a21    a22]

[/br]and

B=[b11  b12][/br]  [b21  b22]

If C is the Matrix product of A and B then:

C=[([a11; a12] dot [b11; b21])   ([a11; a12] dot [b12; b22])][br]  [([a21; a22] dot [b11; b21])   ([a21; a22] dot [b12; b22])]

Notice that an inner product of the rows of A are formed with the columns of B.

The outer product is the opposite of the inner product in the since that the size of the resulting matrix increases instead of decreases. For instance the outer product of the vector [a b] with [c d] is

[ac  ad][/br] [bc bd]

Note that ac denotes a*c

Now lets distinguish between a row and a column vector. A row vector will be a horizontal vecor

For instance [a b]

And a column vector will be a verticale vector:

[c][br][d]

We will use a semi column to denote a new row so [c; d] will also be considered a column vector and [ac ad; bc bd] will be considered a 2 by 2 matrix. In matrix multiplication the inner product of two vectors is:

[a;  b][c  d] or [c;  d][a  b]

And the outer product is:

[a  b][c; d] or [c  d][a;  b]

This is a result of how matrix multiplication is formed by the row by column multiplication.

[googlue 0]

Well, this has started to look like spam.*Gets up to get a broom to clean the mess*If no clear explanations on the posts appear by tomorrow, I will move this to SPAM section

[s243a 0]

To make this more interesting to people who are just taking geometry here is an example of how matrix multiplication can be used for analytic geometry applications. We can represent a point on a plane as so much to the right, say a to the right and so much to the left, say b to the left of a point called the orgin. The orgin could be the bottom left corner of the paper. Thus the other point is:

[a]

 

You can preforme many transformations on this vector for instance you can rotate the point it represents about the orgin by left multiplying the vector by the following rotation matrix

[cos(theta) sin(theta)]

[-sin(theta) cos(theta)]

 

That is if [c; d] is the point after rotation then:

[c] [cos(theta) sin(theta)][a]

[d]=[-sin(theta) cos(theta)]

The matrix:

[cos(theta) sin(theta)]

[-sin(theta) cos(theta)]

 

Peformes a linear transformation on the vector [a; b]

[s243a 0]

Well, this has started to look like spam.
*Gets up to get a broom to clean the mess*

If no clear explanations on the posts appear by tomorrow, I will move this to SPAM section

Pardon me? I spent considerable time writing this. I am not affiliated with any of the sights I linked with hyperlinks. The links were chosen to provide the reader with further information beyond what I have written to help fill in the gaps. Is a discussion of high level math or science forbidden here? Must I always talk to the lowest conman denominator? I tried to proved explanations to my original post because people asked questions about it. Otherwise I could just talk completely over most peoples heads and hope someone with a similar interest joins in.

As far as formatting and such I spent a lot of my time trying to make this post as neat as possible. If there was an alternate text attribute to the image take I could try to replace some of the text by pictures to make it neater. However that would mean my document would not be self contained and if any servers went down the post would make no sense at all. If you could guarantee me web space than I would have a safe place to store my images. However I know the nature here is not to guarantee people web space but rather regiment them to post in a regular pattern following a possibly ridged set of guidelines.

BTW what is the preoccupation here with calling everything spam. Isnt span just chain mail and unwanted advertising. This certainly isnt chain mail and what the heck do you think I am advertising????????

[Moneymaker 0]

are you a math teacher....cuz if ur not...you might want to take up that profession.

[s243a 0]

are you a math teacher....cuz if ur not...you might want to take up that profession.

I am a student studding my masters In electrical engineering (control systems). My thesis is about identifying non linear parameters of a motor using a quasi linear kalman filter. I have two undergraduate degrees, one in physics (math minor) and one in electrical engineering. I would enjoy teaching if I had good students but as far as bellow a university level there is not the same potential to make money. I have yet no engineering work experience but hopefully some day I will find a good job. If I didn?t need a bachelors in education to teach math to high school I could consider it for a bit. However that is not the way the system works. I tried tutoring for a bit. I really enjoyed that. If I had spare time in the future I could consider that as a hobby.

[no9t9 0]

i think the problem with your post is that there wasn't really a point to it. I really don't see what you are trying to ask.. Are you asking someone to help you understand tensors? That's the only quesetion I could get from your post

[s243a 0]

i think the problem with your post is that there wasn't really a point to it. I really don't see what you are trying to ask.. Are you asking someone to help you understand tensors? That's the only quesetion I could get from your post

I think the idea is to share my idea of generalizing matrix multiplication and to get some feedback back. I would like to know are my ideas sound. I would also be interested in learning more about tensors. I may read the link I posted about tensors to help me learn more. I would be interested in how my ideas of generalized matrix multiplication fit into the ideas of tensors. Anyway I just wanted to start some discussion about mathematics here. Someone may bring up another topic and the thread my diverge. Isn?t part of discussion gropus about sharing ideas and learning for each other. For instance in the http://newmars.com/forums/ group I frequently post to, it is not unordinary for someone to propose an idea for space propulsion. The members may then respond by saying how reasonable or unreasonable the idea is and how it fits into present ideas about propulsion. I think sharing information and leaning form each other is an important part of discussion groups. So basically I started off by just sharing some of my currents thoughts on math. Has anyone here been thinking about any mathematical concepts lately?

[OpaQue 15]

Matrices are a Great deal of help.. Atleast in Boolean Algebra where this stuff is really used! Getting the order and type of the sets ... Discrete Structure ( the whole subject ) could had been so much complicated! Matrices really ease the job! As for Tensors, I have studied matrices but I havent hit this word. I know the following concepts related to Matrices,1> Multiplication ( which is row to column one )2> Adjacent Matrix3> normal Matrix4> transpose5> complement6> conjugate7> skew matrix8> Hermatian skew matrix9> Inverse of a matrix10> General Matrix properties & types also the properties related to Matrix Determinants! ( this thing is assumed though )Being a Computer Engineering Student, I havent gone deep into this topic. But I gues tensors are really advance topics of matrices.There are many other things about matrix, but they aint clicking me at the moment.. but if you get an idea regarding my knowledge from the above points, feel free to continue this post ahead! May be my knowledge about Matrix is still in the beginner level.

[Becca 0]

Im like thirteen aswell...[like that other person yeh] and I feel really dumb not knowing what you know...The only things I have recognised so far are sin cos....and I know tan...But thats it...

[s243a 0]

We all have are things we are good at and not good at. We become good at things mostly by hard work but natural ability also helps. If you are only thirteen and already know what sine and cosine are, you are probably ahead of most kids your age. I have been in university for eight years. Each year of university you learn a lot more then in high school and your level increases substantially. I am impressed to even see a thirteen year old try to read what I posted. I purposely made the post really dense with concepts so anyone who was up to the challenge of learning it would learn a lot really quickly. We improve the fastest when we are challenged just enough to push are limits but not enough to exceed them. I feel my greatest gains in conceptual ability come when I feel like I have to beat my head against the desk to figure out the problem or understand something. A good challenge can make the smartest person feel dumb but when they succeed they should remember what they conquered.BTW there seems to be a lot of young people here. I am 26.P.S. If you want to lean about a particular math concept feel free to ask.If you lean this stuff at 13 you will be so far ahead of people it won?t be funny.

[s243a 0]

Thinking back to when I first leaned about sine and cosine I remember it is a way to get the angle in a right angle triangle. Sine and cosine are periodic functions. Many angles can be derived analytically using the unit circle and trigonometric identities. Others must be derived numerically. One way to do this is a series expansion of the trigonometric functions. For instance a Taylor series. To fully understand Taylor series requires a knowledge of calculus.

 

Hmmmm.Locking back maybe I should find another site that I can link concepts to wolframs mathworld has just about everything but some times I think a simpler explanation might be better for some people.