Q:

Understanding the format of a matrix

I know this is a super basic question, but I'm getting mixed up. When creating a matrix, is each row a different vector or each column? In other words, is the first column all the $x_1$ coefficients or is the first row all the $x_1$ coefficients etc?

A:

As https://math.stackexchange.com/users/187663/mathcounterexamples-net points out, first index is the row index, second one is the column. That is the convention used almost everywhere.
However, both the set of rows and the set of columns can both be seen as sets of vectors. This is important, once you start seeing matrices as representing linear transformations, instead of just as little rectangular tables of numbers. We talk about the vector space spanned by the row vectors, and the vector space spanned by the column vectors. Things like that.

