Matrices and linear transformations.

[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."