Are a row vector and a column vector the same thing? Suppose I have 
$$ A = 
\begin{bmatrix}
a\\b\\c\\d\\
\end{bmatrix}$$
$$ B = 
\begin{bmatrix}
a& b& c & d\\
\end{bmatrix}$$
Now, I know $A = B^T$. But in what sense are these different mathematical objects? 
$$
\begin{bmatrix}
1\\
2\\
3\\
4\\
\end{bmatrix} + \begin{bmatrix}
1\\
2\\
3\\
4\\
\end{bmatrix}= \begin{bmatrix}
2\\
4\\
6\\
8\\
\end{bmatrix}
$$
$$
\begin{bmatrix}
1&2&3&4\\
\end{bmatrix} + \begin{bmatrix}
1&2&3&4\\
\end{bmatrix}= \begin{bmatrix}
2&4&6&8\\
\end{bmatrix}
$$
To me, these seem to behave the same way. Is there any difference between a row vector and a column vector? How are $A$ and $B$ different? 
 A: These two representations do work in the same way. The set of all row vectors over a giving ring, $R$, is isomorphic to the set of all column vectors over the same ring, $C$, via the linear transformation  $\phi: R \rightarrow C$ defined by $\phi (v)=v^T$. You can check that this is a bijection and preserves scalar multiplication since you already proved that they behave the same under addition. 
But I will add that often vectors of a given space are represented as column vectors as convention. 
A: Taken by themselves, they are isomorphic to each other. However, in matrices, the rows and columns specify two different vector spaces. There are special relationships between row spaces and column spaces of a matrix but they are different ways of specifying a space or linear transformation. See here.
A: Once you declare that you are dealing with matrices, then they must be different: one is $1$ x $4$ and the other is $4$ x $1$. But there is an obvious isomorphism between them: $[a, b, c, d]\rightarrow 
\begin{bmatrix}
a\\b\\c\\d\\
\end{bmatrix}$
