Studying Linear Algebra: Rn vs. Rm

[zenith92]

Hello, so I'm currently busy my (first) linear algebra course. We use the book Linear Algebra and it's Application third edition Update (red cover). I noticed that in sections 1.4 and section 1.5 (don't know about the rest yet) that they sometimes describe vectors that are in Rⁿ and sometimes in R^m, now I don't know if there's supposed to be a difference or if they just randomly switch between these two letters. I couldn't think of any reasoning behind the different uses, so maybe someone here knows?

Example: A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0, where A is an m x n matrix and 0 is the zero vector in R^m. Such a system Ax = 0 always has at least one solution, namely x = 0 (the zero vector in Rⁿ).

 

[AlephZero]

If you have an equation like
A x = b
where A is an m x n matrix (in other words, A has m rows and n columns),
then x must be a vector with n elements and b must be a vector with m elements. Otherwise, the dimensions are not compatible and the matrix multiplication doesn't make any sense.
That is what "x is in R^m and 0 is in Rⁿ" means.

 

[zenith92]

If you have an equation like
A x = b
where A is an m x n matrix (in other words, A has m rows and n columns),
then x must be a vector with n elements and b must be a vector with m elements. Otherwise, the dimensions are not compatible and the matrix multiplication doesn't make any sense.
That is what "x is in R^m and 0 is in Rⁿ" means.

Oh wow, makes much more sense now thanks a lot. Now I have to go re-read some stuff, though haha