I have been told in the textbook I'm following that "By convention, always assume that vectors are in column orientation, unless stated otherwise." (Cohen, 2021: 27). Fair enough. But as I progress into the study of vector spaces,this distinction between row and column vectors seems to take on more significance.
In particular,I am introduced to fields and vector spaces and being told that "The space $\mathbb{R}^n$ consists of all column vectors with $n$ components". So is this different from having your space made of all row vectors with n components?
Generally: what is the geometrical interpretation of the distinction between row vectors and column vectors? (As I understand it, algebraically, vectors are just ordered list of numbers. The different between row vectors and column vectors here only shows up in the outcome of basic vectors operations like the dot product.)
Best Answer
It is important to not lose sight of what the component representations mean. We start with a vector space $V$. If $V$ is finite dimensional, we can take an ordered basis $(e_i)_{i = 1}^n$. For any vector $v \in V$ we may then uniquely write $v = \sum_i v^i e_i$. We say that the $v^i$ are the components of $v$ in the basis $(e_i)$. We traditionally represent the $v^i$ as a column vector $$ \begin{bmatrix} v^1 \\ \vdots \\ v^n \end{bmatrix} $$ Notice that I say we traditionally do this. It is of course arbitrary how we decide to represent the components of $v$. We could have done it with a row vector or whatever else. But now suppose we want to talk about the components of a linear transformation $A : V \to V$. We can uniquely write $$ Ae_i = \sum_j A_i^j e_j $$ We call the $A_i^j$ the components of $A$. Let's look at what happens when we act on $v$ with $A$. We calculate \begin{align} Av &= A \left( \sum_i v^i e_i \right) = \sum_i v^i Ae_i \\&= \sum_i v^i \left( \sum_j A_i^j e_j \right) \\&= \sum_j \left(\sum_i A_i^j v^i \right) e_j \end{align} We read off from this that the components of $Av$ are $\sum_i A_i^j v^i$. This is the usual rule for matrix multiplication.
Note that I have positioned indices so that we only ever sum over lower and upper pairs of indices. This might give us the inspired choice to represent matrices in $2$ dimensions. We'll say that upper indices run top to bottom in columns and lower indices run left to right in rows. So we represent $A$ with $$ \begin{bmatrix} A_1^1 & \ldots & A_n^1 \\ \vdots & \ddots &\vdots \\ A_1^n & \ldots & A_n^n \end{bmatrix} $$ We see that insisting on this slightly (although not really as you progress in your mathematical career) unusual notation has the benefit of automatically reproducing the "row by column" matrix multiplication rule.
Now what kind of a thing is a row vector? Well really we shouldn't call it a row "vector" at all. It is, as we shall see, more properly called a row co-vector. A co-vector is a linear map $f : V \to \mathbb{F}$, where $\mathbb{F}$ is the underlying field for the vector space. Let's see if we can decide whether a co-vector should have upper or lower indices. We calculate $$ f(v) = f\left(\sum_i v^i e_i \right) = \sum_i v^i f(e_i) $$ We see that $f$ is determined by the values $f(e_i) \in \mathbb{F}$. These values occur in a sum with an upper index. It stands to reason that we should say the components of $f$ are $f_i = f(e_i)$ with a lower index. According to our convention, then, we should represent $f$ as $$ \begin{bmatrix} f_1 & \ldots & f_n \end{bmatrix} $$ We see that co-vectors are what we previously called row vectors. Indeed $$ f(v) = \begin{bmatrix} f_1 & \ldots & f_n \end{bmatrix} \begin{bmatrix} v^1 \\ \vdots \\ v^n \end{bmatrix} $$ which is again consistent with the "row by column" multiplication rule.
Hopefully you can see that these are two very different objects. A column vector is a true vector, whereas a row vector is something that eats a vector and spits out a scalar.
But maybe you think we can just "flip" a row vector to obtain a vector. Of course we could, but then we would lose all the lovely and illuminating structure we've developed so far. We would just be forgetting the distinction between vectors and linear functions of vectors, which I hope you'll agree could be a worthwhile distinction.
If you're curious, a metric is the secret ingredient that lets us translate between row and column vectors. We often tacitly assume a metric that makes this translation the trivial "flip," which leads to a lot of confusion. So I emphasize again: row vectors and column vectors are naturally distinct. A column vector is a vector. A row vector is a co-vector.