Row Vectors vs. Column Vectors - What's the difference?

[Kolmin]

That’s an old time question that it’s still a mistery to me. It’s a lot of time that I am trying to find an answer, but no text is clear on the topic and I am basically self-taught.

What’s the difference between row vectors and column vectors?

I came to this question when I found that the gradient was defined in two different ways on two different books. This was a problem and I started to look around: the more I was searching, the more it became a mistery, cause lot of books state that the gradient is the row vector of the first partial derivatives of a given function.

I fixed this problem in the end (the gradient is not the row vector, but the column vector), but still I don’t get what’s the difference between row and columns, beyond a practical one in terms of computation.

Does exist a "deep" theoretical difference between those two types of vectors or it's a metter of distinction between places (row vectors) and displacements (column vectors)?

Thanks in advance!

 

[Office_Shredder]

There's no real difference. Vectors in your original vector space are typically thought of as column vectors simply so the calculation Ax for a matrix A is a linear transformation from your vector space to your vector space. If you want to do a linear transformation from V to R, (say you want to take an arbitrary vector x and take the dot product with the gradient of a function, which I will call g) then to be able to write this as gx you need g to be a row vector, which is probably why the one book defined the gradient as a row vector

 

[HallsofIvy]

It is fairly common to represent your vectors as columns then you could represent your "co-vectors" (members of the dual space, the space of linear functionals that take each vector to a number) as a row so that the operation of the functional on vector becomes a matrix product:
\begin{bmatrix}a & b & c\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= ax+ by+ cz.

Of course, that is, as Office Shredder says, purely arbitrary- you could always represent the vectors by rows, the funtionals by columns and do the product the other way around.

 

[AlephZero]

It is fairly common to represent your vectors as columns then you could represent your "co-vectors" (members of the dual space, the space of linear functionals that take each vector to a number) as a row so that the operation of the functional on vector becomes a matrix product:
\begin{bmatrix}a & b & c\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= ax+ by+ cz.

Of course, that is, as Office Shredder says, purely arbitrary- you could always represent the vectors by rows, the funtionals by columns and do the product the other way around.

But you do need to be clear about which way round you (or a textbook) IS doing it.
\begin{bmatrix}a & b & c\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} is a scalar, but
\begin{bmatrix}a \\ b \\ c\end{bmatrix}\begin{bmatrix}x & y & z\end{bmatrix} is a 3x3 matrix with rank 1. In some applications (e.g. optimization) both of these are used frequently!

 

[Kolmin]

Thanks a lot.

I kinda had the feeling at a certain point that it was arbitrary, but I found a book that gave a sort of hint about a deep reason behind the use of one or the other. Interestingly enough, this deep reason never showed up, leaving me with nothing more than this doubt.

 

[HallsofIvy]

I suspect that the "deep reason", at least the reason for distinguishing between "row" and "column" was, as I said, to be able to differentiate between "vectors" and "co-vectors" and treat their interaction as a matrix multiplication.