Are column vectors considered the "default" orientation in linear algebra? I am taking an online course where vectors are typically written out as column vectors. It seems like the only row vectors we have seen are the transposes of column vectors and labeled as such.
So I'm wondering if mathematicians (at least those in linear algebra) tend to favor column vectors. This is essentially a question about convention, i.e. is it such a strong convention that if you told a mathematician about a vector without specifying its alignment, would they assume it is a column vector?
Come to think of it I guess that might make sense since it is the first dimension.
 A: This is slighty opinion based, I guess. However, I would say that I've seen column vectors as standard far more than row vectors. So yes, if nothing is stated I would assume column vector.
A: It's possible you haven't seen matrix multiplication defined yet. If that's the case, then this won't be much of a justification.
If $v$ is a row vector of length $n$ (so a $1 \times n$ matrix), and $M$ is an $n \times n$ matrix, then $Mv$ isn't defined although $vM$ would be (and gives back a row vector of length $n$): compare
$$\underset{``\text{nonsense''}}{\pmatrix{1 & 0 \\ 0 &1} \pmatrix{1 & 0}} \quad \text{versus} \quad \underset{``\text{less common, but fine''}}{\pmatrix{1 & 0}\pmatrix{1 & 0 \\ 0 &1}} \quad \text{versus} \quad \underset{``\text{the gold standard''}}{\pmatrix{1 & 0 \\ 0 &1} \pmatrix{1 \\ 0}.}$$
Because we like to think of matrices as functions from a vector space to itself, we probably want to emulate function notation (as in, $f(x)$) and apply matrices "on the left," as in $Mv$ rather than $vM$. Because of the way matrix multiplication is defined, this forces us to use column vectors.
It is much more common in general to see matrices act (from the left!) on column vectors (although there are niche "act on the right" markets).
