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.

• Yes. This is so, when you want to act on a column vector with a matrix, the matrix goes on the left, the same way the name of a function $f$ goes on the left of its input $x$ when you write $f(x)$. – Qiaochu Yuan Feb 20 '17 at 3:00
• Yes it is more or less accepted convention to use column vectors. I have seen some (old) books favor row vectors over column but it seems main reason for it being to save page space and convenience while printing. – Amey Deshpande Feb 20 '17 at 4:06

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).

• (the OP should probably just ignore this comment) It is annoying, though, when you want to talk about the action of a group of matrices on a coordinate vector space, and then, rather confusingly, $$A\cdot(x,y) = A\begin{pmatrix} x \\ y\end{pmatrix}.$$ I suppose one could fix this by talking about actions on the right and multiplying on the right, instead of the left, but then you're going against two conventions. – Will R Feb 20 '17 at 3:06
• @WillR I feel like by the time you're talking about group actions you've internalized matrix algebra well enough that this probably wouldn't throw you, even if it does look strange the first few times. – user137731 Feb 20 '17 at 3:10
• I learned algebra from those in the niche "on the right" market; it took me an embarrassingly long time to figure out why permutation matrices weren't working like they should, on basis vectors -- because I'd learned the version that acts on the right, and had only used them amongst themselves, never on vectors! – pjs36 Feb 20 '17 at 3:19

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.