# Domain of a linear operator

• Oct 6th 2010, 12:09 PM
coley0412
Domain of a linear operator
Does anyone know how to find the domain of a matrix?

Ex:

B=[1,0,1;5,4,9;2,4,6]

I've tried to find a method but I don't find much on the subject.

Thanks!
• Oct 6th 2010, 01:31 PM
Ackbeet
The domain of an $m\times n$ matrix is the set of all $n\times p$ matrices, for all positive integers $p$. The only requirement is that the matrix multiplication be defined. The condition I've listed is the requirement necessary.
• Oct 6th 2010, 01:42 PM
coley0412
My apologies, but would you mind elaborating a little on that? If I have a 2x3 matrix I understand the domain is 3 x p but I don't think that answers the question entirely.

Thanks for the help!
• Oct 6th 2010, 04:28 PM
Ackbeet
Well, there's not a whole lot to elaborate on. You're considering the linear operator consisting of matrix multiplication on the left (that is, the operator matrix is on the left of its operand and multiplies it using regular matrix multiplication). The natural domain of a function or operator is the set of all things that you can "plug in" to the operator. For these operators, the domain is what I've described above.

Is there some more specific portion of this for which you'd like more info?
• Oct 7th 2010, 05:44 AM
coley0412
Well my specific question is what exactly does the answer look like if I'm asked to give the domain of the matrix given in the OP? I do better with examples so I apologize if that's what you've tried to describe in words.
• Oct 7th 2010, 06:46 AM
Ackbeet
A domain is a set, so I would use set builder notation. So, you've got something like

$\displaystyle \mathcal{D}(B)=\{A:A\in \mathbb{R}^{n\times p}\}.$