Why can't we add a non-square matrix $A$ to its transpose $A^T$? The addition operation is commonly defined as follows:

Two matrices must have an equal number of rows and columns to be added

But this is a very shallow definition/interpretation.
A deeper interpretation of a $n \times n$ matrix, I would believe, would be that it is an element of a vector space, e.g. $A \in \mathbb{R}^{n \times n}$ as you need $n^2$ linear equations to compute each entry in the matrix (a dot product for each entry)
Now if you use that interpretation of a matrix, then let's assume we have a column vector $B \in \mathbb{R}^{n \times 1}$ and it's transpose, a row vector, $B^T \in \mathbb{R}^{1 \times n}$
$$\begin{align}B = \begin{bmatrix} 
b_1 \\
b_2\\
. \\
.\\
.\\
b_n\end{bmatrix}
&&\text{and}&&B^T = \begin{bmatrix} 
b_1 &
b_2&
. &
.&
.&
b_n\end{bmatrix}
\end{align}$$
But both $B$ and $B^T$, are both elements of the same vector space $B, B^T \in \mathbb{R}^n$. So why is their addition undefined?
Why can't you add a matrix to its transpose, just like you could two vectors that are elements of the same vector space?
Is it wrong to interpret matrices as I have done? Is there a more rigorous definition of the addition operation for matrices? Are there better ways to interpret matrices?
 A: $B$ and $B^T$ are not elements of the same vector space: $B$ is an element of $\Bbb R^{n\times 1}$, and $B^T$ is an element of $\Bbb R^{1\times n}$. Both of these vector spaces are isomorphic to $\Bbb R^n$, but no two  of $\Bbb R^n$, $\Bbb R^{n\times 1}$, and $\Bbb R^{1\times n}$ are actually equal to each other.
A: I disagree that your proposed definition is "deeper". To me, it is more appropriate to think of $m\times n$ matrices at linear maps $\Bbb R^n\to\Bbb R^m$. From this perspective it is obvious why an $m\times n$ matrix can only be added to a $p\times q$ matrix when $m=p$ and $n=q$. Indeed, a linear map $\Bbb R^n\to\Bbb R^m$ can only be added to another linear map $\Bbb R^q\to\Bbb R^p$ if $n=q$ and $m=p$.
A: To add a layer of interpretation, an $n \times 1$ is often seen as a point in $\Bbb R^n$. A $1 \times n$ matrix will, in the same view, be a linear map $\Bbb R^n \to \Bbb R$. You may apply functions to points (by matrix multiplication), but you do not add functions and points.
A: There is no one deciding what you can and can't do in mathematics. You can do whatever you want to do. The only catch is that you have to convince people that what you do makes sense and is interesting. If you want to define the addition of your $B$ and your $B^T$, go ahead. I'm sure it can be useful in some cases.
