Matrices and linear transformations.

[Studiot]

Thank you all.
I can see, from the confusion expressed by several, that my initial statement may have been taken to read that I propose either that all matrices represent linear transformations or that I wanted to create a vectors tructure.

Neither were the case.

I apologise if I inadvertantly created a false impression, but I thought I has specified matters pretty clearly.

@Dickfore

Thank you for your thoughts.

So, it has been shown that this algebraic structure is not a linear space. Therefore, it does not support linear algebra.

Yes indeed.

However it is representable by matrices.

I guess you/he are/is trying to define multiplication of a vector (in the Hilbert space {red,green,blue}) by a scalar.

Why guess?

Why not just read the definition I gave?

I accept that if I wanted to create a vector space I would have to impose different criteria/structure. But I don't.
Further my scheme is capable of further development.

@Robert

Thank you for your further thoughts.
You still have not indicated what colours are possible for a monocoloured ball.

@TrickyDicky

Thank you for your support.
This thread was created because an apparently very competent mathematician was adament, several times, that all matrices represent linear transformations, full stop. I did not want to disturb the other thread (referenced) with the discussion.

 

[Robert1986]

Then, what kind of structure do they form. I am certainly not aware of such a structure.


Ok, that makes sense.




What does he want to do, exactly?!

He gave an example of a certain type of matrix multiplication of specific types of matrices, such that this matrix multiplication was not just a matrix representation of a linear transformation. I think his initial intent was to provide a counter example to the claim "all matrices are linear transformations" - which he did. However, as micromass pointed out, this has nothing to do with ordinary linear algebra (which is probably the context in which the claim was first made.)

 

[I like Serena]

I believe the point of the thread has already been addressed.
But I find it interesting to ramble on a bit.
Just like the others here apparently. ;)

I've been trying to deduce a algebraic structure here that is consistent and complete.
I'm concluding that the actual dataset of colors is $\{nocolor, red, green, blue\}$.
Let's call it $C$.

$C$ supports a partial operation $+$, which only defines addition of $nocolor$ with any of the other colors.
Note that in algebra and model theory, the definition of an operation includes that all combinations should be defined, which is not the case here.

Furthermore, $1 \cdot color=color$ and $0 \cdot color=nocolor$ (as mm said).

Then the matrix multiplication would be a map $F_2^{1 \times 3} \times C^3 \to C$.
The only reason that this is not a linear map, is because $+$ is not defined for all combinations of elements in $C$.
This restricts the allowed matrices to the ones that have at most one $1$ in them.
So there are 4 allowed matrices.

 

[micromass]

Then, what kind of structure do they form. I am certainly not aware of such a structure.

They form just a set.

What he wants is just to define an algebraic structure where you can do something like

$$\left(\begin{array}{cc} 0 & 1\end{array}\right)\left(\begin{array}{c} red\\ green\end{array}\right)=green$$

So we want a structure that is strong enough such that things like

$$0\cdot red+1\cdot green=green$$

make sense. There are multiple ways to formalize such a structure, but it should be intuitively clear what it does. Clear is that it has little to do with vector spaces or usual linear algebra.

Developing such a system might be interesting in programming and it would not be hard to implement it. However, I don't see much mathematical uses for it.

 

[Dickfore]

Well, I guess you can never be wrong if you are intentionally too vague.

 

[micromass]

Well, I guess you can never be wrong if you are intentionally too vague.

Was that directed to me?? I agree that I might be too vague, but why do you think it was intentional?

 

[Dickfore]

Was that directed to me??

No, it was directed to the OP. I only saw there were two new posts after I had posted.

 

[TrickyDicky]

Then, what kind of structure do they form. I am certainly not aware of such a structure.

Developing such a system might be interesting in programming and it would not be hard to implement it. However, I don't see much mathematical uses for it.

Look at differential geometry and what chiro said in post #5.

 

[Dickfore]

What is this
$$\left[\begin{array}{c} \mathrm{blue} \\ \mathrm{red} \\ \mathrm{green} \end{array}\right] + \left[\begin{array}{c} \mathrm{green} \\ \mathrm{blue} \\ \mathrm{red} \end{array}\right] = ?$$
equal to?

 

[micromass]

What is this
$$\left[\begin{array}{c} \mathrm{blue} \\ \mathrm{red} \\ \mathrm{green} \end{array}\right] + \left[\begin{array}{c} \mathrm{green} \\ \mathrm{blue} \\ \mathrm{red} \end{array}\right] = ?$$
equal to?

I don't think studiot wants to define addition of matrices.

 

[I like Serena]

What is this
$$\left[\begin{array}{c} \mathrm{blue} \\ \mathrm{red} \\ \mathrm{green} \end{array}\right] + \left[\begin{array}{c} \mathrm{green} \\ \mathrm{blue} \\ \mathrm{red} \end{array}\right] = ?$$
equal to?

It is undefined, since + on real colors is undefined.

However, making things consistent and complete, we might say:
$$\left[\begin{array}{c} \mathrm{nocolor} \\ \mathrm{nocolor} \\ \mathrm{green} \end{array}\right] + \left[\begin{array}{c} \mathrm{nocolor} \\ \mathrm{blue} \\ \mathrm{nocolor} \end{array}\right] = \left[\begin{array}{c} \mathrm{nocolor} \\ \mathrm{blue} \\ \mathrm{green} \end{array}\right]$$

 

[TrickyDicky]

What is this
$$\left[\begin{array}{c} \mathrm{blue} \\ \mathrm{red} \\ \mathrm{green} \end{array}\right] + \left[\begin{array}{c} \mathrm{green} \\ \mathrm{blue} \\ \mathrm{red} \end{array}\right] = ?$$
equal to?

If studiot said explicitly he wasn't defining a vector space , why would you think that addition should be defined?

 

[I like Serena]

If studiot said explicitly he wasn't defining a vector space , why would you think that addition should be defined?

Indeed Studiot never defined addition.
It's what we are inferring to fit the matrix multiplication in the framework we are used to.
And indeed, it is possible.

 

[Dickfore]

If studiot said explicitly he wasn't defining a vector space , why would you think that addition should be defined?

http://en.wikipedia.org/wiki/Matrix_(mathematics)

Matrices of the same size can be added or subtracted element by element.

 

[Dickfore]

Also, what is the meaning of:
$$\left[\begin{array}{ccc} 0 & 1 & 1 \end{array}\right] \, \left[\begin{array}{c} \mathrm{blue} \\ \mathrm{red} \\ \mathrm{green} \end{array}\right] = ?$$

 

[micromass]

Also, what is the meaning of:
$$\left[\begin{array}{ccc} 0 & 1 & 1 \end{array}\right] \, \left[\begin{array}{c} \mathrm{blue} \\ \mathrm{red} \\ \mathrm{green} \end{array}\right] = ?$$

Such a multiplication would not be defined. The only multiplication that would be defined is if the left matrix has exactly one 1 on each row.

 

[I like Serena]

http://en.wikipedia.org/wiki/Matrix_(mathematics)

That article does not say that addition has to be defined.
A matrix is just a rectangular set of symbols.
Furthermore, it allows to impose more mathematical structure to it, but that is not part of the definition of a matrix.

Also, what is the meaning of:
$$\left[\begin{array}{ccc} 0 & 1 & 1 \end{array}\right] \, \left[\begin{array}{c} \mathrm{blue} \\ \mathrm{red} \\ \mathrm{green} \end{array}\right] = ?$$

Explicitly undefined.

 

[Dickfore]

So, nothing is defined. What we have are "red", "green", and "blue" by themselves. No need of matrices here.

 

[TrickyDicky]

http://en.wikipedia.org/wiki/Matrix_(mathematics)

I'm not sure if that requirement is already referring to linear algebra matrices, but in any case it would be up to Studiot to define the result of that addition. I don't think he is going to develope a whole consistent non-linear system just to make his point which is already clear.

 

[micromass]

So, nothing is defined. What we have are "red", "green", and "blue" by themselves. No need of matrices here.

Well, I certainly agree that there is no need of matrices here. But apparently studiot wanted them for some reason.

 

[Dickfore]

But, they are not matrices, since they don't fit the usual definition of matrices. They are some weird notation of a row containing only a single one (not a row vector1) that picks out a "red", "green", and "blue" from another column array (not a column vector!) by some weird "rule" that 'Studiot' thought fitting to write.

 

[I like Serena]

The matrix is designed as a notation to "pick" a color from an ordered set.

Studiot's point/question was that not every matrix defines a linear transformation.
And that is true, when you take it out of the context of linear algebra.

 

[micromass]

But, they are not matrices, since they don't fit the usual definition of matrices. They are some weird notation of a row containing only a single one (not a row vector1) that picks out a "red", "green", and "blue" from another column array (not a column vector!) by some weird "rule" that 'Studiot' thought fitting to write.

Well, I guess this depends on your definition of matrix. In my point of view, a matrix is just a rectangular array with certain entries. In that respect, he certainly did define matrices. But they are kind of useless since almost no operation is defined.

 

[TrickyDicky]

But, they are not matrices, since they don't fit the usual definition of matrices. They are some weird notation of a row containing only a single one (not a row vector1) that picks out a "red", "green", and "blue" from another column array (not a column vector!) by some weird "rule" that 'Studiot' thought fitting to write.

I already gave an example of a nonlinear matrix in #25, the 2X2 invertible complex matrix, to which you can associate a Mobius transformation, you can make all kinds of computations with it, and they fit the definition of matrix.

 

[Dickfore]

Well, I guess this depends on your definition of matrix. In my point of view, a matrix is just a rectangular array with certain entries. In that respect, he certainly did define matrices. But they are kind of useless since almost no operation is defined.

Well, your definition of a matrix is certainly not the one commonly accepted by a math community.

 

[I like Serena]

Well, your definition of a matrix is certainly not the one commonly accepted by a math community.

Well, if you look up "matrix" on wiki, first you get a disambiguation of about 40 options.
And if you pick "matrix (mathematics)", it says:

In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries.​

Furthermore:

Matrices of the same size can be added or subtracted element by element.​

Note the use of "can".
I find the article a bit sloppy in the sense that it almost seems to imply that all mathematical operations and even a specific multiplication type should be defined on it.

 

[Dickfore]

According to what rule in the definition of a matrix are the following forbidden:
$$\left[\begin{array}{ccc} 0 & 1 & 1 \end{array}\right], \ \left[\begin{array}{ccc} 1 & 0 & 1 \end{array}\right], \ \left[\begin{array}{ccc} 1 & 1 & 0 \end{array}\right], \ \left[\begin{array}{ccc} 1 & 1 & 1 \end{array}\right]$$?

 

[I like Serena]

According to what rule in the definition of a matrix are the following forbidden:
$$\left[\begin{array}{ccc} 0 & 1 & 1 \end{array}\right], \ \left[\begin{array}{ccc} 1 & 0 & 1 \end{array}\right], \ \left[\begin{array}{ccc} 1 & 1 & 0 \end{array}\right], \ \left[\begin{array}{ccc} 1 & 1 & 1 \end{array}\right]$$?

In the definition of a matrix they are not forbidden.
With the extra restrictions that have been set for this problem, they are not allowed.
Or perhaps we could say that you can use them, but then the result is undefined.
Similar to $\left[\begin{array}{ccc} 0 & 0.1 & 1 \end{array}\right]$ as not being allowed.

 

[Dickfore]

In the definition of a matrix they are not forbidden.
With the extra restrictions that have been set for this problem, they are not allowed.
Similar to $\left[\begin{array}{ccc} 0 & 0.1 & 1 \end{array}\right]$ as not being allowed.

No, it's not similar. We have established that the left matrix can take values from the set $\left\lbrace 0, 1 \right\rbrace$. In your example 0.1 does not belong to the set. So, his "restrictions" contradict the definition of a matrix. Therefore, it is not a matrix.

 

[micromass]

Well, your definition of a matrix is certainly not the one commonly accepted by a math community.

Can you give me a definition that is commonly accept by the math community? The notion of "matrix" seems to be a little like the notion of "number", it is undefined but everybody knows what it means.

 

[TrickyDicky]

Actually I am not 100% sure the objects defined by Studiot in the OP are rectangular arrays, as opposed to vectors-like objects, I would have to have that confirmed by a mathematician.
But the fact he might have chosen a not completely perfect example doesn't mean he is wrong about what he was trying to clarify.

 

[Dickfore]

Can you give me a definition that is commonly accept by the math community? The notion of "matrix" seems to be a little like the notion of "number", it is undefined but everybody knows what it means.

See the posts before yours.

 

[micromass]

Therefore, it is not a matrix.

Well, please define matrix.

 

[Dickfore]

Actually I am not 100% sure the objects defined by Studiot in the OP are rectangular arrays, as opposed to vectors, I would have to have that confirmed by a mathematician.
But the fact he might have chosed a not completely perfect example doesn't mean he is wrong about what he was trying to clarify.

Yes, it does.

 

[I like Serena]

No, it's not similar. We have established that the left matrix can take values from the set $\left\lbrace 0, 1 \right\rbrace$. In your example 0.1 does not belong to the set. So, his "restrictions" contradict the definition of a matrix. Therefore, it is not a matrix.

The allowed matrices are defined as having entries from {0,1} and having only a single 1.
I have to admit that the last condition was added in a later post.

So the allowed matrices are a subset of $F_2^{1 \times 3}$.

My example violates the first condition, whereas yours violates the second condition.

Edit: All of the examples are still conform the wiki definition of a matrix in mathematics.