Matrices and linear transformations.

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Discussion Overview

This thread explores the proposition that all matrices define linear transformations, examining the relationship between matrices, linear maps, and their contexts, including potential exceptions. The discussion includes theoretical considerations, mathematical reasoning, and conceptual clarifications.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that every m x n matrix defines a linear transformation through left-multiplication, while others question this by introducing contexts where matrices may not correspond to linear maps.
  • A participant suggests that the colors used in an example can be treated as labels and questions the necessity of relating them to R^n.
  • There is a discussion about the nature of the set {0,1} and its classification as a field, with some participants asserting it is a field while others express uncertainty.
  • One participant proposes that if colors are treated as independent attributes, they can form a basis in a vector space, while another challenges this by suggesting that not all matrices correspond to linear transformations.
  • Concerns are raised about the implications of defining operations in contexts that may not adhere to the properties of vector spaces, particularly regarding scalar multiplication and the definition of linear transformations.
  • Participants discuss the distinction between input and output datasets in the context of a matrix equation, emphasizing the need for clarity in definitions.

Areas of Agreement / Disagreement

Participants express differing views on whether all matrices define linear transformations, with some asserting this as a general principle while others contend that certain matrices do not fit this definition. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are limitations regarding the definitions of fields and vector spaces, as well as the assumptions made about the independence of colors and their representation as vectors. The discussion also touches on the implications of extending mathematical definitions beyond their physical interpretations.

  • #61
Dickfore said:
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.
 
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  • #62
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 />?
 
  • #63
Dickfore said:
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.
 
  • #64
I like Serena said:
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.
 
  • #65
Dickfore said:
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.
 
  • #66
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.
 
  • #67
micromass said:
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.
 
  • #68
Dickfore said:
Therefore, it is not a matrix.

Well, please define matrix.
 
  • #69
TrickyDicky said:
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.
 
  • #70
Dickfore said:
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.
 
  • #71
micromass said:
Well, please define matrix.

Is 1 a number, if 1 + 1 is not defined?
 
  • #72
Dickfore said:
Yes, it does.

I refer you again to my example then.
 
  • #73
Dickfore said:
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
 
  • #75
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.
 
  • #76
Dickfore said:
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.
 
  • #77
Dickfore said:
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.
 
  • #78
Dickfore said:
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...
 
  • #79
Dickfore said:
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. ;)
 
  • #80
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.
 
  • #81
Well I certainly have made folks think.

:biggrin:

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.
 
  • #82
I like Serena said:
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.
 
  • #83
Studiot said:
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.
 
  • #84
Studiot said:
Well I certainly have made folks think.

:biggrin:

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).
 

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