Matrices and linear transformations.

[micromass]

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

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:
<br /> \left[\begin{array}{ccc}<br /> 0 &amp; 1 &amp; 1<br /> \end{array}\right] \, \left[\begin{array}{c}<br /> \mathrm{blue} \\<br /> \mathrm{red} \\<br /> \mathrm{green}<br /> \end{array}\right] = ?<br />

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:
<br /> \left[\begin{array}{ccc}<br /> 0 &amp; 1 &amp; 1<br /> \end{array}\right], \ \left[\begin{array}{ccc}<br /> 1 &amp; 0 &amp; 1<br /> \end{array}\right], \ \left[\begin{array}{ccc}<br /> 1 &amp; 1 &amp; 0<br /> \end{array}\right], \ \left[\begin{array}{ccc}<br /> 1 &amp; 1 &amp; 1<br /> \end{array}\right]<br />?

 

[I like Serena]

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

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.

 

[Dickfore]

Well, please define matrix.

Is 1 a number, if 1 + 1 is not defined?

 

[TrickyDicky]

Yes, it does.

I refer you again to my example then.

 

[micromass]

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.

So a (0,1)-matrix is not a matrix? http://en.wikipedia.org/wiki/(0,1)-matrix

 

[micromass]

Is 1 a number, if 1 + 1 is not defined?

Why should elements of matrices be numbers? Again, see (0,1)-matrices: http://en.wikipedia.org/wiki/(0,1)-matrix

 

[Dickfore]

Since we have managed to stray in the field of arbitrariness of definitions, and are not willing to accept the other party's arguments, I decided to back away from this thread.

 

[I like Serena]

Is 1 a number, if 1 + 1 is not defined?

No, not necessarily.
For instance, in abstract algebra {1,2} is a group with multiplication modulo 3.
In particular 1+1 is not defined.

 

[TrickyDicky]

Since we have managed to stray in the field of arbitrariness of definitions, and are not willing to accept the other party's arguments, I decided to back away from this thread.

I accepted your arguments, but you didn't even acknowledge mine once.

 

[micromass]

Since we have managed to stray in the field of arbitrariness of definitions, and are not willing to accept the other party's arguments, I decided to back away from this thread.

Well, the problem seems to be that you never provided a definition of a matrix...

 

[I like Serena]

Since we have managed to stray in the field of arbitrariness of definitions, and are not willing to accept the other party's arguments, I decided to back away from this thread.

Definitions in math are not arbitrary.
To the contrary, they are very sharply defined.
To understand what those definitions are exactly, is now what this whole thread is about.

But I can certainly understand that you had enough of it. ;)

 

[TrickyDicky]

Dickfore, I'd bet you are not really agreeing with "the proposition that all matrices define linear transformations" that the OP was trying to prove wrong, regardless of how fortunate his example was.

 

[Studiot]

Well I certainly have made folks think.

However, I don't see much mathematical uses for it.

The first matrix is extendible. I have only shown one row but you could have many rows. In my example this would correspond to many trials of ball withdrawal. However less trivial results might be a connectivity diagram for an electrical network or structural framework.

 

[micromass]

Definitions in math are not arbitrary.
To the contrary, they are very sharply defined.

I don't think I agree. For example, the notion of "number" does not seem to have a good definition in mathematics. Should complex numbers be numbers? p-adic numbers? transfinite numbers? I don't know any standard definition of number.

 

[micromass]

The first matrix is extendible. I have only shown one row but you could have many rows. In my example this would correspond to many trials of ball withdrawal. However less trivial results might be a connectivity diagram for an electrical network or structural framework.

Yes, boolean matrices (which are similar) are already used in electical networks. But there you specifically use the structure of boolean algebras.

 

[TrickyDicky]

Well I certainly have made folks think.

Thanks, it's been fun, and we might have set up some record for brief and fast posting not counting the non-science subforums (almost 60 posts in a little over 2 hours).

 

[TrickyDicky]

Also:

http://en.wikipedia.org/wiki/Transformation_matrix

"Linear transformations are not the only ones that can be represented by matrices."