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

[AndreTheGiant]

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?

Homework Equations

The Attempt at a Solution

 

[Mark44]

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.

 

[AndreTheGiant]

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?

 

[Mark44]

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.

If i set X = (xij) and Y = (yij).
What are those?

That notation is usually used for matrices, where x_ij represents an arbitrary element of the matrix. Same for y_ij.

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.