Understanding Vector Spaces: The Confusion of Notation in Coefficient Matrices

[stgermaine]

If I am given a coefficient matrix of m rows and n columns (an m x n matrix), then is the vector space of that matrix Rm or Rn? I get really confused sometimes. Sometimes, the superscript seems like the number of rows, and sometimes the number of variables. It also doesn't help that my class textbook uses the notation n x m whereas the Lay Linear Algebra textbook, which is far superior to the one used in my class, uses m x n.

Most of the time, it seems like it corresponds to the number of variables, so the number of columns in a coefficient matrix, so an m x n matrix is a vector space of Rn.

 

[NegativeDept]

If I am given a coefficient matrix of m rows and n columns (an m x n matrix), then is the vector space of that matrix Rm or Rn?

I find it helpful to use an informal definition of "vector space":

A vector space is a set of things and some rules for making linear combinations of those things.

$\mathbb{R}^m$ and $\mathbb{R}^n$ are two different vector spaces. Suppose we choose a standard basis for each of those spaces, and we agree to represent vectors as columns of their components using those bases. If $\hat{L}$ is a linear transformation that inputs vectors from $\mathbb{R}^n$ and outputs vectors in $\mathbb{R}^m$, then we can represent $\hat{L}$ as an $m \times n$ matrix. (It's easy to get $m$ and $n$ confused. When in doubt, I pick two small numbers for $m$ and $n$ and write down an example. Then it's usually clear if I've gotten it backwards.)

We can also make linear combinations of matrices: a matrix can be multiplied by a scalar, two matrices can be added, and the rules of * and + are well-behaved. That means the set of all $m \times n$ real matrices forms another vector space. If I remember correctly, this new vector space is isomorphic to $\mathbb{R}^{m n}$.

For example, the set of all $2 \times 3$ real matrices is a real vector space with dimension 6. Each matrix represents a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$. I hope that clarifies things!

 

[HallsofIvy]

Well, first you are going to have to explain what you mean by "the vector space of that matrix"! Of course, a matrix can represent a linear combination from one vector space, U, to another, V. One standard way to do that is by the matrix multiplication Au= v, thinking of u and v as "column matrices" with a single column. If we do that, then the definition of matrix multiplication requires that u have as many rows as A has columns and that v have as many rows as A has rows. That is, if A has "m rows and n columns", u must have n rows, so be in R^n and v must have m rows and so be in R^m. That is, if A has m rows and n columns, A represents a linear transformation from R^n to R^m.

 

[stgermaine]

A quote from my textbook says "Note that BA is an nxm matrix (as it represents a linear transf. from Rm to Rn)

And further on it says that "the eqn z = B(Ax) = (BA)x for all vectors x in Rm. I guess what you are saying about linear tarnsformation makes sense, since a lot of these matrices are being multiplied. I was just wondering in cases when the textbook says stuff like "for all vectors x in R2"

And then it hit me that the number of 'rows' on a column vector correspond to the number of columns in a coefficient matrix. I think that's one place where I got mixed up about what the superscript means in Rn.

 

[dimension10]

It also doesn't help that my class textbook uses the notation n x m whereas the Lay Linear Algebra textbook, which is far superior to the one used in my class, uses m x n.

Depends on whether you're talking about row vectors or column vectors?

P.S. Why is this in homework help?

 

[stgermaine]

@Dimension10 I think I was getting column vectors and row vectors confused. Maybe I should have posted this on Linear & Abstract Algebra.

Anyway thank you all I'm no longer confused about this.

 

[HallsofIvy]

If this was a question about course work, then this is indeed the correct place for it.