Understanding Notation for R^(n x m) and Its Use in Regression Analysis

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AndreTheGiant
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Homework Statement



So yeah as the topic says. What does R^(nxm) mean? Is that a matrix or an n x m space. The notation is really confusing and sometimes it means something else.

I'm doing regression analysis right now and this shows up a lot. Is it a matrix when you say something like C = ℝ^(k x m). I'm not sure if that means its a k x m matrix or a k x m space. Mainly because in examples when they say that, it shows up as a k x m matrix. I'm guessing its a a matrix because of the = sign? If you had an E sign instead it would be contained in a space?

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The Attempt at a Solution

 
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AndreTheGiant said:

Homework Statement



So yeah as the topic says. What does R^(nxm) mean? Is that a matrix or an n x m space. The notation is really confusing and sometimes it means something else.

I'm doing regression analysis right now and this shows up a lot. Is it a matrix when you say something like C = ℝ^(k x m). I'm not sure if that means its a k x m matrix or a k x m space. Mainly because in examples when they say that, it shows up as a k x m matrix. I'm guessing its a a matrix because of the = sign? If you had an E sign instead it would be contained in a space?
It's the vector space of n X m matrices, with real entries.
 
Thanks! That cleared up so much...

Another quick question if you may know.

If i have the observations. (xi1,...xik,yi1,...,yim). If i set X = (xij) and Y = (yij). What are those? Is that an i x k matrix for X and an i x m matrix for Y?
 
AndreTheGiant said:
Thanks! That cleared up so much...

Another quick question if you may know.

If i have the observations. (xi1,...xik,yi1,...,yim).
xi1,...xik would be the i-th row of a matrix with k columns.
yi1,...,yim would be the i-th row of a matrix with m columns.
AndreTheGiant said:
If i set X = (xij) and Y = (yij).
What are those?
That notation is usually used for matrices, where xij represents an arbitrary element of the matrix. Same for yij.
AndreTheGiant said:
Is that an i x k matrix for X and an i x m matrix for Y?
Probably not i x k and i x m, but <something> x k and <something> by m. I can't tell from what you have here.