Probably a rookie question, but here it comes anyway: is there any difference between the "regular" definition of the dot product of two vectors
$$\vec a \cdot \vec b = \sum a_i b_i$$
and the formulation involving the transpose of the first vector?
$$\vec a^T \cdot \vec b = \sum a_i b_i$$
The latter definition makes sense when you think of vectors as just being matrices which only have one column. But in that sense, the first definition is a bit weird, considering the fact that you can't multiply a $3\times 1$ matrix with another $3\times 1$ matrix, which would make the first definition impossible.
So, which definition is "better", if any? Is the first just a sloppy, lazy way of writing things or are both equally true.
But then, how can both be equally true at the same time? If you think of vectors as matrices, the first definition doesn't work. If you think of vectors as their own thing, which are different from matrices, i. e. objects which have a magnitude and a direction, the act of transposing these objects doesn't chance a thing (or does it?), which would make it a bit pointless to define the transpose of a vector just to "save" the second definition of the dot product.
I am a bit confused by this. Especially since some authors use the first definition, while others always bother to write thier dot products in the form $\vec a^T \cdot \vec b$.