Matrix mulitplication in PDF: Is this a typo or am I misunderstanding?

[KindofSlow]

HelloAll,
Thisis from the document found here:
https://math.dartmouth.edu/archive/m12f10/public_html/RepresentingMatrix.pdf

(7) (Very importantexample) Let A be any m x n matrix.
Then A defines alinear map L_A : Rn →Rm
(“left-multiplicationby A”) defined by L_A(v) = Av for v ϵ Rn.

Note that this makessense: the matrix product of A (an n x m matrix) and v (a column vector in Rn,i.e., an n x 1 matrix) is an m x 1 matrix, i.e., a column vector in Rm.

My question:
First it says A is m x nmatrix, then it looks to me like it says A is an n x m matrix.

Is this a typo or amI misunderstanding?

Thank you

 

[ksdhart2]

I agree with your assessment that it's just a typo. Specifically, the second mention of A is the typo, and the actual dimensions of the matrix A are meant to be m by n. You can see that must be the correct interpretation by seeing that if n=2 and m=3, a 2x3 matrix cannot be multiplied by a 2x1 matrix, due to dimension mismatch. However, a 3x2 matrix can be multiplied by a 2x1 matrix, and gives a 3x1 matrix as a result, as specified by the text.

 

[KindofSlow]

Great, thank you.

 

[HallsofIvy]

Frankly, I can never remember whether the "m" in "m by n" refers to the number of column and the "n" to the number of rows or vice-versa!