The meaning of a matrices

Matrices can be considered as a system that describe a set of points in n-dimensional space.

Let's consider these equations


1.)this equation describe a set of points in a 3-D space. More specifically, it appears as a plane.


2.)this equation describe another set of points in a 3-D space too.



3.) this two equations describe a new set of points in space. More specifically, "it" together appears upon us as a line, rather than two isolated planes.


4.) If there is a new equations such that the unknowns can be solved.
Then these equations becomes in a way:

Are those equations same as before? What are the meanings of them?

The answer is no for the latter two. They are NEW sets of points in a 3-D space (geometrically planes) by each of them independently.
All three of them together are described as, merely, ONE point in space.

What I cannot create, What I do not understand

What I cannot create, What I do not understand-Feynman

After learning the concept in a book, you should create the concept in on your own, in other words, try to express it even better than the writer.

It is a strange and beautiful world

I have not yet lost a feeling of wonder, and of delight, that this delicate motion should reside in all the things around us, revealing itself only to him who looks for it. . . . . . To see the world for a moment as something rich and strange is the private reward of many a discovery. -E.M. Purcell

My high school bible teacher often told me how amazing the miracles of God were. However, the things around us everyday are even much more amazing and interesting compared to the miracles.

Try to touch your fingers by another finger of yours. What do you feel? Do you feel something that...... you are touching? What is the force against your fingers such that your fingers won't get through another? It is electric repulsive force. I am always shocked by such a fact!!

Another interesting thing is Color. In fact, Our consciousness on color is dependent upon the wavelength of light. Stuff itself doesn't have such a thing called color. It all takes place just in our brains.

But in fact physics is a most exciting activity. It has to do with understanding, and it is most remarkable in that such great depth of understanding seems to be possible. -P.1 in "mathematical physics-a popular introduction" by Francis Bitter

Doing Sequences in a cool way

Try to solve it without looking downward

1.)

One can think of it till the cows come home or use the formula, but there is another cool way:
check it out:



Obvously the answer will be one over three, one third.

there is another picture:


So there is another question:


2.)

But I can't think of one in a geometrical way.
If you can, please tell me.


and there is another question:


3.)

what is the picture?

pretty obviously, the answer becomes one over eight

ok, let us think of another question:


4.)
I have no idea for the geometrical picture. Or is there such a geometrical picture?

So what is the trick?

Why some of could them be easily found, but some else we hardly had any idea?

what is in commen between 1.) and 3.) ? and what else is in common between 2.) and 4.)?

Matrices, matrices and matrices' dimensions

Another good function of matrices is that it shows its dimensions clearly.

Let there be a 3 x 3 matrices and 5 x 5 matrices, M & W:



M has two vectors and each of it has two unit vectors, and every unit vectors has its unique direction. So a vector with two unit vectors is in a two dimensional space. And a vector with five unit vectors, W, is in a five dimensional space.

Now let's see what property a hype-dimensional vector has:

First, what mean by "vertical" here?

We know it very well in a 3-D space, our space, but we know little in a hyper-dimensional space, so what mean by "vertical" here?

Second, what mean by "slope" here?

and lots else coming too. Think about it when you are free=)

Matrices, matrices and matrices

In Linear Algebra,
the rows of a matrix can be viewed as vectors.

So the interesting things come after such a brilliant idea.

Let's see a 2x2 matrix, M

in Latex: \left(\begin{array}{cc}3 & 2\\2 & -1\end{array}\right)

so there are two vectors <3,2> & <2,-1> in a 2-D space

given
what happens after sigma 1 times M is that M still is M!


So it is reasonable to regard sigma 1 as "one".

Besides, there are many other functional matrices,
Like, such a matrix makes another matrice upsidedown




so an interesting question comes:

WHY and HOW?

宏微之妙

原子奇之又奇, 眾妙之門!
從微到宏, 其妙不可言!
宏中有微, 微中有宏!!