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vit_r
Please ignore IDs -- they are for robots.
Please ignore titles -- they are for me.
Please ignore errors in English for now. It would be (probably) improved.
DW comments are screened and could be not only opened but also deleted.
If you like to start discussions, it is better to do this in your blog. You can reuse IDs and images. It is better if your changes are recognizable, but it does not matter. This is a draft, and I would not explain yet the concepts of extensions.
Please ignore titles -- they are for me.
Please ignore errors in English for now. It would be (probably) improved.
DW comments are screened and could be not only opened but also deleted.
If you like to start discussions, it is better to do this in your blog. You can reuse IDs and images. It is better if your changes are recognizable, but it does not matter. This is a draft, and I would not explain yet the concepts of extensions.
Sometimes you discover that the modern civilization has lost
some pretty obvious knowledge.
You can only wonder.
It is understandable that the modern generations of mathematicians
do not know functional analysis.
Our professor had said
that his bright PhD student
had spent two months to understand the basics of his own mathematical knowledge
(which he had excellently demonstrated many times)
and to get the first real life equation right.
Of course,
this time waste is unacceptable today.
A modern PhD student
feeds data and equations into a computer
and in two months gets almost ready results.
(Note, we live now in the world of almost ready everything.)
This means,
this modern PhD student performs the main task --
to demonstrate achievements --
and nobody bothers,
if his equations are wrong
and if the results the computer had given him, her -- or now it, them, ver, xem, etc. --
is meaningless.
(By the way,
if you value this modern English,
it is better for you to stop and leave here.
You could feel offended for mathematics you know.)
This means,
the functional analysis is rightfully forgotten
because it is obviously ineffective for modern science.
The funny discovery of the December 2021
was the information
that some people do not understand the matrix multiplication
in the way it is explained to the engineers.
You multiply a 2-dimensional matrix A by a 2-dimensional matrix B.
(As a side note:
Your language thinks for you
and it would be better to use the German "to multiple with"
or the Russian "to multiple on",
but we stay with the wrong correct English.)
| .- | -. | .- | -. | ||||||||
| | | a1,1,1 | a1,2,1 | | | | | b1,1,1 | b1,1,2 | b1,1,3 | b1,1,4 | | | ||
| | | a2,1,1 | a2,21 | | | × | | | b1,2,1 | b1,2,2 | b1,2,3 | b1,2,4 | | | |
| | | a3,1,1 | a3,2,1 | | | *- | -* | ||||||
| *- | -* |
The result of this multiplication is a 2-dimensional matrix C.
| .- | -. | |||||
| | | c1,[1,2],1 | c1,[1,2],2 | c1,[1,2],3 | c1,[1,2],4 | | | |
| | | c2,[1,2],1 | c2,[1,2],2 | c2,[1,2],3 | c2,[1,2],4 | | | |
| | | c3,[1,2],1 | c3,[1,2],2 | c3,[1,2],3 | c3,[1,2],4 | | | |
| *- | -* |
The equation is A×B=C.
Anything is good unless you ask, why the cells in the matrix C are calculated in the following way.
c1,1,1
=
(a1,1,1
×
b1,1,1)
+
(a1,2,1
×
b1,2,1)
c2,1,1 = (a2,1,1 × b1,1,1) + (a2,2,1 × b1,2,1)
...
c2,1,1 = (a2,1,1 × b1,1,1) + (a2,2,1 × b1,2,1)
...
The correct answer is: Because a result of a multiplication of two 2-dimensional matrices
is 3-dimensional,
but it is too complicated to work with 3-dimensional matrices.
The 3-dimensional multiplication is (or at least should be) obvious for an engineer
and the explanations are banal,
but it happens even
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golos_dobra
does not understand
how engineers explain the matrix multiplication.
This explains
why his illustrations are lacking.
Below I correct some of them.
golos_dobra
does not understand
how engineers explain the matrix multiplication.
This explains
why his illustrations are lacking.
Below I correct some of them.
Note, there is a fundamental difference between "to know" and "to understand".
The modern civilization turns from the civilization of understanding
into a civilization of knowledge
(or more precise: into a civilization of googled knowledge).
There is also a fundamental difference between reading how to think
and training oneself in the engineering thinking.
At the longest night of the year 2 c.d. (COVID domini)
I had drawn some explanation images for my son
because this theme would be good to learn
some aspects of the engineering thinking.
The following text contains these images with some additional explanations.
I do not expect that everybody would understand
them.
I could prepare a perfect textbook for children
which even a modern mathematician
with the modern flat thinking
could understand
but it would be too boring to write.
By the way,
it would be not interesting to spend time
for converting the notes for my son into something presentable,
but I was interested in understanding
how to generate SVG images
and this theme was a good for a throw-away prototype.
This means,
all images below were generated.
Some of them contain manually added decorations.
The software developers create software to play with it,
but at some point you should stop.
It was too time consuming to write code for generation of additional decorations
and too boring to modify other images manually.
If you like,
you could print them and draw on them the additions
that could help you to obtain better understanding.
If you scan the results and then destroy the paper,
your process would be called "a paperless technology".
(Do not be too romantic and do not let to blind you with cool buzzwords --
they all do the same thing.)
Please,
do not ask about my source code.
If you had read "The Software Dragons",
a throw-away prototype is made
to get the first outcome -- the result,
and the third outcome -- the knowledge.
The source code -- the second outcome of a software project --
is initially not intended to be used anymore.
It is always more effective
to rewrite the first draft completely
on base of the obtained knowledge
and the discovered top-level mistakes
in the software architecture.
However,
today the former throw-away prototype
would be called "a release".
I have now the source code
on the same level,
but in the contrary to the modern development processes
it does what it was intended to do,
and it does not contain errors
which must be found by users.
It is simply a draft.
This text contains keys you can use to find the answers yourself.
I advise to collect them and to think before you read further.
(I do not expect
that everybody could understand these explanations,
but I expect some engineers to learn something new.)
Note,
this is a web post and I -- in comparison to the book pages --
cannot control what you could see.
It is also too boring
to write complete and precise explanations.
If you are interested to know
how engineers think,
I advise you to take a pen and a lot of graph paper (I use 5 mm squares),
and to try to find out the answers before you see them.
I would use color pens
for explanations for children
and for people who do not understand engineering --
for instance, for mathematicians --,
but all images below are for engineers,
engineers can understand black and white.
The first key are the matrices above.
They are enough for advanced engineers.
The next key could be obtained,
if you open the source code and replace
style="opacity: 0.0"
with
style="opacity: 1.0"
I think,
the riddle is too complicated for people
who do not know the true nature of mathematics.
This is an advanced topic
and I have explained
to my son many mathematical principles
which I would not repeat here.
Otherwise,
it would be impossible to follow the explanations
if you do not know the basics.
I shortly mention only two important principles: Lost Dots and Renaming.
Lost dots
The mathematicians are for ages bad at design.
The way they write their formulas and draw illustrations
is a sure way to introduce errors.
If you had a high school education with lectures in advanced mathematics,
you could remember that the most frequent activity of your lector
was the search for a lost minus.
Of course,
the signs "+" and "-" are too difficult to recognize
and a mathematician usually loose a small vertical line
or unconsciously adds it in a wrong place.
Guess, why we write "2ab" when we mean "2 multiplied by a multiplied by b"?
Just because mathematicians use the sign ":" for the division operation.
If they do not recognize a prominent line,
they would permanently mix one dot with two dots.
When the engineers have started to develop computers,
they could recognize this mess
and they have replaced a dot with a star "*".
They even have added a line to the "0" because a zero sign
could be mistaken for letter "O".
Unfortunately,
the time of clarity has now ended,
and the mathematicians active participate now in the development
of computer languages.
Even the old industry proven languages
are now polluted with garbage.
It is responsible for many complicated errors
that are difficult to find,
but it is liked by modern software developers
because they are not engineers
and use the programming languages for self-enjoyment.
I use here the sign × (∧times;)
which is usually used by teaching multiplication
to primary school kids.
Some mathematicians would claim
that this is wrong,
but I would not give a damn.
This sign also represents
a rotated plus sigh,
crossed lines,
four right angles,
and other things that have meaning
from the point of view of an engineer.
I would show things
mathematics usually hides.
This deceives people
who do not know the hidden rules of decoding.
For instance,
you could write:
2ab = 2 × a × b
How about "2cm"?
It could be
2cm = 2 × 10-2 × m
However, it could also be
2cm = 2 × 2. 99792458 × 108 × m/s × y × kg
The letter "c" could denote the standard value for the speed of light and the letter "m" could denote an inertial mass.
And the new "m" is not the former "m", because the previous "m" was for some unknown "mass"
and the new "m" is for distance units called "meters".
Look, I have written an unknown variable as an "y" and not as an "x".
Just because I am not a mathematician but an engineer.
I would not claim that each idiot
could distinguish a sign "×"
and a letter "x".
I simply prevent the source of errors.
Anything in mathematics is dependent on hidden meaning.
You could simplify a formula, if you see a letter "a".
But you could not do the same trick if you see another letter
Where is the difference?
The first version has hidden dots between the letters
and the "d" in the second version is not a parameter but a specific limit of differences: (xn+1-xn).
| ax | x | |
| ---- | = | ---- |
| ay | y |
| dx |
| ---- |
| dy |
However,
you cannot believe in "d".
This is only the fourth letter,
"a", "b", and "c" could be quickly used,
and you must find a dot between "d" and "x"
if you see something like.
10ax3y - 2bx2y2 + 3cxy + 2dx
However,
you must probably not,
if this formula is placed within an integral.
The matrix multiplication is based on hidden indexes.
The second mathematical trick is the deceiving with names.
Renaming
The mathematicians are bad at design,
but they are especially bad at naming.
They could not think out proper names
for the theorems they create
and are forced to name them by their own family name.
When they discover the second theorem
they do not give it a proper name,
they simply add a number.
Of course,
this is a source of many errors.
There are some mathematicians with the same family name,
and they are even not family related.
There are also many cases
when different mathematicians claim to find out the same thing.
This means,
mathematicians could not create clarity
even for the most important achievements of their lives.
Our professor had told us about big name war
between Moscow and Leningrad mathematicians
and had advised us to remember that these names
denote one concept
but from different geographical locations.
If the mathematicians were more inventive,
we could have a "space with collapsing dots",
"mother-in-law teeth" as a class for specific geometrical figures,
a "crossed chain theorem"...
But we have only names, names, names.
I do not like the modern mathematics --
it looks like cemetery.
Do not believe in names the mathematicians give
to objects,
concepts,
theorems
and even the variables in your tasks.
The hidden renaming is another trick of the modern mathematics.
If you are weak in such coding and decoding,
you could not get descent marks from your teacher.
I would yet not explain the renaming in detail from the psychological point of view.
Let's consider a simple illustration of this mathematical trick on a simplest case.
We have the following formula.
x2 - 2abx + a2b2
It is obvious that we can apply the simplest mathematical transformations.
a2b2 => (ab)2
2abx => 2xab
We can apply a lot of other transformations,
but this way leads to the right goal.
Others only waste time.
Why? Because we unconsciously apply the following renaming.
2abx => 2xab
x => a
ab => b
Why we apply it? Of course, not while an "x" looks like an "a".
We apply it because we recognize the textbook formula.
ab => b
a2 - 2ab + b2
The students who were not deceived in the previous steps
could easily remember the equation.
(a - b)2 = a2 - 2ab + b2
Now the solution is simple:
We apply the previous formula backwards and then apply the unconscious renaming backwards.
a => x
b => ab
Then we write the answer hiding all previous steps because they are "obvious".
b => ab
x2 - 2abx + a2b2 =
(x - ab)2
This is how the school textbooks work.
A teacher could also add a remark that only an idiot could not solve a such simple task.
(It is always a simple trick to solve a task you give to a student,
if you already know not only the solution
but also have seen how this task is solved by another student.)
If you mix many tasks with dubious renaming,
a lot of students could make
mistakes what gives you a nice distribution of marks for your exams.
As you can see, the modern mathematics has nothing to do with real life.
The most important rule which any student must know and apply first and foremost is:
It means not how it seems.
Simply speaking,
the mathematics is a fraud,
and it is good for a cognitive psychologist
to study modern schoolbooks to learn perfect methods of laying and deceiving.
(We do this with my son.)
The illustrations are not sufficient,
and you need a full course of cognitive psychology
to understand the true face of mathematics,
but I stop here and return to matrices.
Matrix engineering
A matrix is an ordered storage of cells
where you could store everything.
The kind of order is not important,
the trick is to find the cell by the references
to store some content or to extract it.
The content of a cell is also not important.
You could store in a matrix cell
a banana,
a magnetic field,
a letter,
or a number.
Everything means everything as long as this is something abstract
and could be processed as a mathematical object.
Teachers like numbers
because they give the feeling of knowledge.
I would not use them
because this feeling is deceiving.
If you multiply a banana on a magnetic field
you get the modern science.
This is not interesting
as long as you do not have a scientific grand for such crap.
I would use length as an engineering object
which is mathematically correct
and cognitive straightforward.
If you add a length to a length,
you get a length.
If you multiply a length by a length
you get an area.
From the engineering point of view,
we have multiplied a 1-dimensional object
by a 1-dimensional object,
and we have got a 2-dimensional object.
What would happen,
if we place these lengths in three matrix cells?
Sorry, the sample NsF::20211222::2020 is too obvious
and I place it below the multiplication of two 2-dimensional matrices.
Try to draw the multiplication
of 2 scalars
as if they were 0-dimensional matrices
A and B which contain only one cell each.
I use a rectangle as a representation of a cell
and usually place the cells connected to build a rectangular table.
However,
it is also reasonable to cut such tables to separate cells.
A matrix can be many-dimensional (0, 1 and 2 are included in "many").
The cell of a matrix would be defined by 0, 1, 2, or more indexes.
The mathematicians claim
that the 0-dimensional matrix should be called a scalar,
the 1-dimensional matrix should be called a vector,
the 2-dimensional matrix have the right to be called a matrix,
but if a matrix has 3 or more dimensions
it should be called a tensor.
Our professor had advised to use the Pot Rule:
"You could call it even a pot as long as you do not place it in an oven."
I use the term "matrix"
and you could always say
that a scalar c is a cell c1,1,1,1,1,....
You could add any number of indexes
if your only have the first element by this index.
Note,
this was one of the most important keys.
Let's try the matrix multiplication.
We start with the simplest case.
I multiply the left matrix A with 3 cells ai
by a top scalar b.
The result is obvious:
a multiplication of a 1-dimensional matrix with 1-dimensional
length in each cell by a 1-dimensional length
produces a 1-dimensional matrix with 2-dimensional areas in each cell.
By the way,
it is boring to write obvious explanations.
The following notes will be shorter.
Let's place the length b
in a matrix B
with one cell b1.
(Note,
I have used here a deceiving mathematical notation.
The index on the image NsF::20211222::2101 is correct
and is also a next key.)
Obvious, correct and boring.
Next sample is a multiplication of a 2-dimensional matrix
by a scalar.
Yet gets a bit interesting.
We place the scalar b in the same a matrix B.
What says mathematics?
If you ask a mathematician
where the problem is,
you get an explanation
that you are not permitted to do something
that contradicts the rules of matrix multiplication.
You even get the rules cited.
If you repeat the "Why?" question
and ask about the reasons of the rules,
you usually fall into a cyclic discussion.
By the way,
the trick is simple.
There is a matrix 1.
(Mathematicians tent to use special UNICODE characters
which look like a number 1 but are a bit thicker,
or which some funny decoration that is usually not recognizable.)
You always could multiply
on the 1 between other mathematical operations
because this is a special case of multiplication
which does not change the result.
This means,
there are many different ways
to turn a scalar into a matrix.
| b × | [ 1 ] | = | [ b ] |
| b × | [ 1 , 1 ] | = | [ b , b ] |
| b × | [ 1 , 1 , 1 ] | = | [ b , b , b ] |
| [ 1 , 1 ] | [ b , b ] | ||
| b × | [ 1 , 1 ] | = | [ b , b ] |
| [ 1 , 1 ] | [ b , b ] |
Note,
the representation of a 2-dimensional matrix as a column of rows is correct.
This would be explained below.
The sample NsF::20211222::2120 is too obvious
and is moved down.
Try to draw the right matrix 1
and to get the same result as in the multiplication
of a 2-dimensional matrix by a scalar. (NsF::20211222::2110)
Now I multiply a left 1-dimensional matrix A with cells ai
by a top 1-dimensional matrix B with cells bj
and get a 2-dimensional matrix C with cells ci,j.
Anything works as expected.
Don't worry!
This is a degenerate case.
Note,
the indexes on the image NsF::20211222::2210 are correct,
the paragraph above it contains the deceiving mathematical index placement.
Let's change the directions of matrices A and B.
What is the mathematical answer?
A mathematician could say
that you could not multiple
a matrix l × m
by a matrix n × o,
you can only multiply
a matrix i × k
by a matrix k × j.
Yes,
i × k × k × j
is the key.
If you ask a mathematician,
what are the i, j, and k,
the answer would be, they are dimensions.
This is right.
If you ask what for dimensions are they,
the answer would probably be
that they are the dimensions of the matrices
you are permitted to multiply.
This answer is mathematically correct.
This means,
it is useless and deceiving.
Let's reduce the top matrix B to 3 cells.
We repeat the multiplication.
It is now permitted
but the result is funny.
The matrix C is not a matrix,
but a scalar.
And this scalar is calculated by dubious rules.
How it looks?
It looks like a fraud.
Yes, it is.
If you are an engineer,
you could find the rules
and correct the image.
This is the sample NsF::20211222::2240 below.
Finally,
I give you the last set of keys.
Please take a sheet of paper -- and I really mean this --
and draw on it the first sample of the multiplication
with the left matrix A 3 × 2,
the top matrix B 2 × 4,
and the resulting matrix C 3 × 4.
I have cut the left matrix in 2 columns
and the top matrix in 2 rows,
but this is not important.
Dimensions are indexes.
Indexes are axes.
Please draw axes i, j and k
above top left edge of the matrices A, B, C.
If you did not get the trick,
slowly scroll to the sample
NsF::20211222::2250.
If you cannot recognize it,
you are not qualified as an engineer.
Note,
forget about matrices.
To free yourself from mathematical thinking
you must see the image as an image.
I hope,
now you can show with your hands
how the matrix multiplication works.
By the way,
the hands are an excellent tool for mathematical explanations,
but if you try to explain the matrix multiplication
to a child or to a mathematician,
you could draw the matrices A and B on paper strips,
make to cuts above and on the left of the matrix C
to rotate the matrices A and B
and to put them down through the cuts.
One immediate step.
It is not needed
but here is the right place to show the multiplication of a length by a length in a matrix representation.
The matrices A and B are 2-dimensional matrices with one cell.
What is the matrix C?
If you are explaining the matrix multiplication to a child
or to a mathematician,
you could cut out this matrix multiplication and to assemble it.
Next image represents the matrix multiplication not as a projection on 3 dimensions
but as a 3-dimensional image.
The fractalization index k creates a new dimension.
Case closed.
However,
it remains a funny part:
the conversion of an engineering matrix multiplication
into a mathematical matrix multiplication.
This is a master class of deceiving.
First we cut the resulting 3-dimensional matrix C
into 3-dimensional columns
on the 2-dimensional field.
Note,
the illustrations of the matrix C
as an object divided into vertical 3-dimensional columns (NsF::20211222::2253),
as an object divided in layers (NsF::20211222::2251),
and even a projection from above (NsF::20211222::2250)
are different only in perception.
They are not distinguishable
from the mathematical point of view.
It does not matter,
how you place the cells on your illustration
and which real world analogies you find for them.
These are not the changes of meaning,
these are the changes in perception.
Here is an explanation you would remember.
The same process
could be psychologically perceived quite differently
by different people.
What one perceives as a rape,
that another perceives as an adventure.
The German history of the 21st century is full of adventures.
A lot of adventures await ahead.
(I have read
the New Year article and the comments on Reitschuster.de.)
It seems,
just now we have lost some mathematicians.
This does not matter,
let's see how a matrix multiplication 3 × 1 × 1 × 3 turns into a scalar.
I have changed the lengths in a,1 and b,3 to improve visibility of different layers.
Now,
the matrix C is a 3-dimensional column with 3 layers.
To make mathematics,
we return it in 2 dimensions.
This is pretty simple:
you can place in a cell anything,
and this means you can place in a cell of a matrix
a complete another matrix.
I draw a 2-dimensional matrix D with the single cell d1 ,1.
This cell still has the indexes i and j.
This cell has no layers,
this means it does not have the index k.
Of course,
we can find the objects in layers c1,1,1, 1,2,1, and 1,3,1
by finding the cell d1 ,1 by defining i=1 and j=1.
and then by referencing the content within this cell by the index k.
Matrix operations are performed in parallel.
It does not matter,
what order of operations do you use,
as long as the result is the same.
Yet we apply mathematics.
Now,
if you ask a mathematician,
what is inside the cell d1 ,1,
the answer would be: "An area, another area, and just another area".
The operators "," and ", and" are usually replaced with a "+".
If you add a banana with a magnetic field,
you get the modern education,
but the mathematicians usually use numbers.
This means,
a mathematician could write:
d1,1 = an area + another area + just another area
Of course,
a mathematician writhes this a bit different:
d1,1 = a1b1 + a2b2 + a3b3
A matrix with one cell looks like scalar.
The letter "e" is reserved for exponents,
the letter "f" is used for functions.
This means,
a mathematician reuses the letter "c":
c = d1,1
c = a1b1 + a2b2 + a3b3
The last mathematical operation is to forget all intermediate steps
because they are "obvious".
c = a1b1 + a2b2 + a3b3
Last sample is the multiplication of
a 2-dimensional matrix A by a scalar b.
The equation is
A × 1 × b = C
Where the matrix 1 is
| [ 1 ] |
| [ 1 ] |
Note,
the matrix C has the same dimensions as the matrix A.
You can rotate the squares that represent cells in any direction.
These are only simple representations of abstract objects.
Flat squares were used only to make matrix dimensions recognizable
and to make obvious
the connections with the illustrations we usually see.
What is the hidden meaning of the matrix 1?
This is the dimension transformation matrix.
Let's consider the space with dimensions i, k, j
The simplest matrix 1 would copy a flat matrix A
into a 3-dimensional matrix C which looks like a copy of the matrix A:
The matrix C would be a 3-dimensional matrix,
which has values only on its diagonal cross-section.
| [ 1 , 0 , 0 ] |
| [ 0 , 1 , 0 ] |
| [ 0 , 0 , 1 ] |
If could rotate the cross-section with values
by slightly changing the matrix 1.
| [ 0 , 0 , 1 ] |
| [ 0 , 1 , 0 ] |
| [ 1 , 0 , 0 ] |
You could try to write the usual flat indexes i and j
and to use matrices 1 to transform the matrices on the page
into matrices in a 3-dimensional multiplication.
Yes,
I could do this in case I would develop the dynamic illustrations.
This is also a key
in the world of the multiplication of 3-dimensional matrices.
Two final notes
First and foremost, do not ask Who?, When? and How?
I do not know.
It would be nice to say
that 30 years ago
our heigh school lector had shown us
the matrix multiplication
in this way.
It may be true,
but most mathematicians are like dogs:
You feel,
they know something,
but they cannot explain it,
they can only show.
It would be nice to say
that I had immediately recognized the 3-dimensional nature
of the 2-dimensional matrix multiplication.
I was trained in engineering thinking,
I had learnt Fortran,
I had started to study C,
I could understand the matrix multiplication in the way I have demonstrated above.
At least, the basics
because now after 30 years of engineering
I understand quite different almost everything.
I could remember,
if I could use matrix multiplication.
Of course,
I have intensively used matrices in different programming languages,
but I have stored there: test results, personal data, function references,
matrices,
matrices of matrices,
etc.
I did not use matrices for complex calculations.
The most mathematical task was the transformation of graphical objects.
I also do not know,
why and how this information was lost.
The mathematicians are not to blame.
They are forced to save space.
If you have 210 pages of calculations
and you are forced to limit the size of your article,
you obviously throw out the "obvious".
Even the dots take space.
The same limitation is applicable for the textbook writing.
You need 5 data slots for a 1-dimensional vector with the length 5,
25 data slots for a 2-dimensional matrix 5 × 5,
and 125 slots for a 3-dimensional matrix 5 × 5 × 5.
Of course,
you would save dimensions.
Especially when these are some numbers on paper.
The early computers could preserve the fractalization dimensions,
but the computer memory was too expensive.
It was economically correct to make slightly more intensive calculations
even on the ancient slow processors
to save the limited computer memory.
The software developers of the past were forced
to strip the years from first 2 digits
because this could save space.
The modern software is based on old algorithms,
and I do not know,
if the 3-dimensional results could be at any real use
by a complex parallel processing.
As I nave mentioned,
I did not have real tasks with matrix calculations.
Yes,
this is a good visualization,
and it would reduce mistakes by solving heigh school tasks.
If you get a real-life task,
you usually could reuse some already prepared algorithms.
Of course,
good explanations are not welcomed by teachers.
This makes an important part of their work meaningless.
You could not spend a semester for a simple theme,
if your students could completely understand it in the first lecture.
Note,
the 3-dimensional matrix multiplication is not the most important knowledge
that is already lost by the modern education system.
The modern students have severe problems
with the most basic concepts and skills.
Second,
do not expect "the same but for 3 and more dimensions".
Most people fail in understanding 3-dimensional objects.
I have lost too much time for these explanations,
and I do not know how many people could understand them.
According to the joke of golos_dobra
I would need a head position in an academic department
(Mathematics is useless, I mean the cognitive psychology).
It would be needed an indefinite budget for computer simulation
and a lot of PhD students
eager to work for unclear reasons.
Maybe in 10 years we could understand
how to explain a rotation of a 3-dimensional object
in a 5-dimensional space.
golos_dobra
I would need a head position in an academic department
(Mathematics is useless, I mean the cognitive psychology).
It would be needed an indefinite budget for computer simulation
and a lot of PhD students
eager to work for unclear reasons.
Maybe in 10 years we could understand
how to explain a rotation of a 3-dimensional object
in a 5-dimensional space.
Note,
the verbs "to show" and "to explain" have quite different meaning.
Especially,
if we consider the modern flat-thinking generations of the smartphone-kids.
Advanced tasks
NsF::20211222::1301
Take any sample above and transform it into an explanation in the style
which is usually used in mathematical lectures and articles.
- ::a Write down an educational explanation for heigh school students.
- ::b Write down a scientific explanation which is mathematically correct.
NsF::20211222::1202
- ::a
Draw a 12-sector ring matrix A --
this is a matrix, that is rolled around a circle.
The index of this matrix has the following sequence of values:
[ "Aries"
, "Taurus"
, "Gemini"
, "Cancer"
, "Leo"
, "Virgo"
, "Libra"
, "Scorpio"
, "Sagittarius"
, "Capricorn"
, "Aquarius"
, "Pisces"
].
Note, these index values are not better and not worse than decimal values [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] or hexadecimal values [ #x01, #x02, #x03, #x04, #x05, #x06, #x07, #x08, #x09, #x0A, #x0B, #x0C ]. The important part is only the access to an indexed cell.Multiply this ring matrix A by a one-dimensional matrix B (1 × 2).Note, the result is not a cylinder.
- ::b Imagine, you are a mathematician. Flatten the result on the plane in the most obscure way and find the most obscure rules for such multiplication.
Final remarks
The images are made with the slab serif font Vaccine designed by Manvel Shmanovnyan and released
by ParaType, Inc..
Please ignore the previous line.
This is simply a command for robots to prevent idiots
from finding this page.
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no subject
Date: 2022-01-03 10:09 pm (UTC)очень круто.
no subject
Date: 2022-01-04 01:31 am (UTC)В целом мне хватило одной картинки и физического смысла что трехмерная матрица это набор трансформаций, отсюда идет суммирование или проецирование как финальная операция.
вообще жаль, что крайне редко дают такие простые физичные метафоры математических операций
по поводу умножения помня что i*j j*k размерности, чтобы было умножабельно придется либо транспонировать кольцо и тогда получится цилиндр либо рассматривать зодиаки как 1*1 и тогда получится два кольца
no subject
Date: 2022-01-04 08:06 am (UTC)Я сейчас смотрю учебники начальной школы. Всё даётся так криво, что даже родители с высшем образованием не всегда понимают, что учителя от детей хотят получить. Это не говоря про велосипедистов, крутящих педали по 34 часа в день или развивающих скорость больше 150 километров в час.
no subject
Date: 2022-01-04 09:02 am (UTC)Да, въ математикѣ много разныхъ исторически сложившихся обозначенiй, которые надо запомнить и не путать. Но эти обозначенiя мѣшаютъ только начинающимъ школьникамъ и студентамъ. Главная сложность математики не въ этихъ обозначенiяхъ, а въ томъ, чтобы научиться работать съ произвольными, только что опредѣленными символами и абстракцiями.
Кромѣ того, математика на уровнѣ выше школьнаго и университетскаго уже не оперируетъ такими обозначенiями, какъ d/dx. Тамъ все гораздо болѣе логично и болѣе сложно.
Вотъ, кстати, есть такая книга, гдѣ авторы разработали новыя обозначенiя и въ нихъ переписали теормеханику. "Structure and interpretation of classical mechanics".
https://www.amazon.de/Structure-Interpretation-Classical-Mechanics-Sussman/dp/0262194554
https://www.amazon.de/Structure-Interpretation-Classical-Mechanics-Press-dp-0262028964/dp/0262028964/ref=dp_ob_title_bk
https://www.fisica.net/mecanicaclassica/struture_and_interpretation_of_classical_mechanics_by_gerald_jay_sussman.pdf
Цитата изъ предисловiя:
The traditional use of ambiguous notation is convenient in simple situations, but in more complicated situations it can be a serious handicap to clear reasoning. In order that the reasoning be clear and unambiguous, we have adopted a more precise mathematical notation. Our notation is functional and follows that of modern mathematical presentations.
...
We require that our mathematical notations be explicit and precise enough so that they can be interpreted automatically, as by a computer.
...
When we started we expected that using this approach to formulate mechanics would be easy. We quickly learned though that there were many things we thought we understood that we did not in fact understand. ... The resulting struggle to make the mathematics precise, yet clear and computationally effective, lasted far longer than we anticipated. ... We hope others, especially our competitors, will adopt these methods that enhance understanding, while slowing research.
no subject
Date: 2022-01-04 09:54 am (UTC)Когда любая тема рассматривается с более высокого уровня, все предыдущие шаги кажутся элементарными. (Чаще всего неправильно.)
Короче, это вопрос, мучающий известного персонажа: "Зачем читать детектив, если можно посмотреть на последнюю страницу и узнать, что убийца -- садовник?"
А когнитивная сложность проблематична в двух, даже, в трёх аспектах: 1) бесполезно усложняет вхождение в тему, 2) приводит к ошибкам в использовании, 3) ограничивает возможности двигаться дальше и глубже от заученного.
no subject
Date: 2022-01-04 11:26 am (UTC)Я категорически не согласенъ, что это "boring" - записывать и мотивировать во всѣхъ деталяхъ сначала простые, потомъ чуть болѣе сложные примѣры и т.д. Это какъ разъ самое важное. Только такъ можно научить новой интуицiи и понять, откуда въ данной наукѣ ноги растутъ.
Я считаю, что записывать и обосновывать надо всѣ детали. Нужно дать ученику небольшую, на замкнутую область знанiя, въ которой все ясно и мы ограничены только временемъ и силой нашихъ мозговъ. Потомъ въ этой области знанiя можно прорѣшать много примѣровъ и наработать интуицiю. Я считаю, что только на основѣ полнаго овладѣнiя деталями можно получить интуицiю и двигаться "дальше и глубже". Нельзя давать сначала какую-то общую высокоуровневую картину, это будетъ впустую потраченное время.
Напримѣръ, разсказъ про матрицы я бы начиналъ съ того, зачѣмъ нужно вообще умножать матрицы. На этотъ вопросъ по существу отвѣтить нельзя, если считать, что матрица - это "прямоугольная таблица изъ чиселъ". Прямоугольную таблицу изъ чиселъ можно заключать въ рамочку и вѣшать на стѣну или показывать начальству. Надо говорить не о прямоугольныхъ, треугольныхъ или пятиугольныхъ таблицахъ, а спросить, какiя практическiя задачи рѣшаются съ помощью умноженiя матрицъ. Если ученикъ еще не понимаетъ даже самой постановки такихъ задачъ, то говорить о матрицахъ вообще рано.
no subject
Date: 2022-01-04 01:24 pm (UTC)no subject
Date: 2022-01-14 07:04 am (UTC)no subject
Date: 2022-01-14 07:39 am (UTC)no subject
Date: 2022-01-04 08:50 pm (UTC)>какiя практическiя задачи рѣшаются съ помощью умноженiя матрицъ.
намного хуже - тут нужно объяснять, зачем нужно понимать, как перемножаются матрицы. потому что окей, такая-то задача решается перемножением матриц. пусть матлаб или numpy нам их перемножает, если вдруг попалась такая задача. мне лично совершенно не очевидно, что это умножение нужно как-то понимать для решения практических задач.
no subject
Date: 2022-01-04 02:03 pm (UTC)Это вам не квадратные трехчлены представлять!
no subject
Date: 2022-01-04 03:06 pm (UTC)no subject
Date: 2022-01-04 03:28 pm (UTC)Математически там два кольца и это не интересно. Практически применяют совершенно другую модель.
no subject
Date: 2022-01-05 11:43 am (UTC)no subject
Date: 2022-01-06 05:44 am (UTC)Bозник вопрос, почему при умножении скаляров, остаемся в двумерном пространстве, а при умножении матриц снова складываемся в двумерное?
no subject
Date: 2022-01-06 08:24 am (UTC)Например, умножение длин -- это переход из одного измерения в два. Если считать длины а и b, то получим двумерный объект ab. Если считать 5 м на 6 м, то получим 30 м2, потому что числа перемножаются и удобнее работать с результатом, чем таскать дальше 5*6. Также как и ab могли положить в какой-нибудь c.
Что, кстати, при преобразованиях тех же уравнений часто приходится делать, чтобы не разносить формулы на три листа, а держать их компактно.
Надо понимать, что эта тема рассмотрена после многих других. До двух листов с матрицами идёт ещё кипа тех, где "очевидное" объясняется, начиная с самых основ. Естественно, здесь это пропущено. Если начать приводить записи в товарный вид, это работа на несколько месяцев.
no subject
Date: 2022-01-12 04:32 am (UTC)По моему, проще к матрицам подходить со стороны решения системы линейных уравнений.
Потом преобразование координат.
Затем через линейную регрессию и анализ многомерных данных.
no subject
Date: 2022-01-12 09:53 am (UTC)А математика так сконструирована, чтобы было очень сложно. Упрощения и абстракции начинаются только на более высоком уровне. Исторический подход к обучению, будь он неладен.
no subject
Date: 2022-01-11 01:52 pm (UTC)No, it doesn't, a matrix with one cell is not a number. Matrix multiplication is defined for matrices, and produces matrices. And there is another operation defined for matrices, multiplication by a number. These two look the same on a sheet of paper, but they are not, it's a type error.
no subject
Date: 2022-01-13 06:36 pm (UTC)Как-то я пропустил эту длинную телегу. Обсуждать вломак, алгебра есть алгебра, и я как-то не ожидаю тут алгебраических новостей. Или?
no subject
Date: 2022-01-13 06:51 pm (UTC)no subject
Date: 2022-01-13 10:26 pm (UTC)О, занятно.
Мне такие ощущения чужды, так что могу только поцокать языком.