What is the significance of Rn and Rm in Calculus?

[amb123]

I think I'm lost on a key concept in LA. I was asked a question about what MxN a matrix needed to be in order to follow laws for T(x) = Ax to go from R4 to R5. I was completely lost. I think I'm still not sure what it means to be Rn, and if only speaking of one R is it Rn, and it two it must be Rn and Rm. And, what does Rn by itself indicate, and what do Rn and Rm together in a problem indicate?

If this makes no sense, then please help me to clear up my problem. I just had the Ch1 test for this course and no matter the question, the answers are almost always just to put the second vector into a matrix with the first vector or first matrix and find if there are free variables, or prove consistency. Beyond that, I'm having a lot of trouble understanding a lot of the terms they throw around in the questions (realm, span, what T(x) really is, etc, etc). I did well in Calculus, but I'm not seeing this as easily.

Thanks for any guidance, and yes, I have read the book, again, and again, and again.

-A

 

[HallsofIvy]

Rⁿ (n-dimensional Euclidean space) can be thought of a n-points ("ordered n-tuples") or as the corresponding vector space.

R² is two-dimensional space which we can think of as points in the plane (x,y) or two dimensional vectors: ai+ bj.

R³ is three-dimensional space which we can think of as points in space (x,y,z) or three dimensional vectors: ai+ bj+ ck.

R⁴ is four-dimensional space, etc.

Whether you think of it as a point-space or a vector space basically depends upon whether you want to be able to do "vector operations" (add vectors, multiply by scalars) or not. Since you are talking about matrix operations you clearly are talking about Rⁿ as vectors.

If you have an "M by N" matrix, then you have a matrix with m rows and n columns.
To multiply that by a single vector, you basically take the "dot product" of each row with the single column representing the vector: Of course, you have to have the same number of components to "match" them up: since the matrix has n columns, each row will contain n numbers and so your vector must have n numbers: it must be in Rⁿ. You do that for each row to get the components (numbers) making up the result: since there are m rows, the result will have m numbers: it must be in R^m.
Seems to me you are having some fundamental problems "internalizing" the definitions. That's a heck of a lot more than we can help with here. I recommend you go talk to your professor and perhaps set up some time for one-on-one work with him/her.

 

[amb123]

Thanks, yes, I am having problems grasping the concepts. The professor is not a permanent instructor, and has no office hours. He has set aside "15 minutes" after each class to help people. Many need help, and this is only a 2 day a week class. He doesn't go over the concepts, for the first hour he reviews homework questions, then the last 15 minutes he assigns homework and does on example from the next section. There haven't really been any good discussions about what this stuff means:(

So, if you have n columns, you must have an Rn vector to multiply it by. What about the original matrix, can it be considered to be an Rn matrix?

I think the question I would really like clearing up on said that following law where T(x) = Ax, what mxn does A need to be in order to map (a vector?) from R4 to R5? (something close to that, does it make any sense?)

If you have some time, I would appreciate more help. Thanks.

 

[mathwonk]

as halls of ivy explained, a matrix multiplies by a vector, by taking the dot product of each row with the given vector. thus the matrix mkust have the same size rows as the vector that you are multiplying by. mkoreover the number of dotproducts you get mout equals the number of rows in your matrix. thus a matrix with 2 rows, each of length 3, i.e. a 2 by 3 matrix, will multiply by vectors of size 3, i.e. by vectors from R3, and will yield 2 answers, i.e. 2 numbers, i.e. a vector in R2.

hence a 2by 3 matrix sends vectors from R3 to vectors in R2.
So what size matrix eould you need to send a vbector from R4 to R5? or from R5 to R4?

do dome research in your book. surely this is in there, under "representing linear maps by matrices" or something like that.

lang's book had section called "the matrix assopciated to a linear map", and then another called "the linear map associated to a matrix".

 

[amb123]

*click*. I am getting it. I started reading ch 2 in the Lay book and it is really making things click. I think that this should've been the first chapter! Much of the class failed the Ch1 test, unfortunately. I got an 83, which I would usually regard as a very bad grade, but seeing as my understanding on much of the material was really poor, an 83 is great.

Your explanation was also helpful, about sending vectors from R2 to R3, etc. I have been having a hard time with terminology, but I'm finally getting it after just reading a bit into Ch2.

Thanks so much!
Angela.