When do you write the vector in a row or column... our book/ teacher seems to jump back and forth...
It is pretty arbitrary but sometimes it is VERY important which one's used. For example, a row vector with n components can been seen as a 1 x n matrix, and the same column vector as a n x 1 matrix ==> you can matrix multiply the former by the latter (in this exact order!), but NOT the other way around unless n = 1.
Don't worry, you'll see this later on.
When you say solve systems of equations you mean Gauss or
Gauss-Jordan reduction, right?
Then you operate on rows, and your aim is to set the system in row-echelon form.
You could although in theory write everything as columns, but it would be
awkward to write an equation like "x+y=4" on it's side.
Now for spans and general vectors it doesn't really matter which way you write it.
The reason you write spans usually with column vectors is that they look
nicer side by side than row vectors (and you will see another reason when working with
linear maps as matrixes) . It is just convention to write it
as column vectors.
Like tonio said before the determinant of a matrix is independant of wether
you write the vectors as columns or vectors.
Your calculus book probarbly writes vectors like rows because you are
used to working with cordinates (x,y) before.
So my point is it really doesn't matter, don't waste too much time thinking about
whether to write rows- or columns . Try to focus on other more important things in
your linear-algebra class.