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

Kolmin · Mar 272012

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 answerbut 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 searchingthe more it became a misterycause 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 vectorbut the column vector)but still I don’t get what’s the difference between row and columnsbeyond 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 · Mar 272012

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 functionwhich I will call g) then to be able to write this as gx you need g to be a row vectorwhich is probably why the one book defined the gradient as a row vector

HallsofIvy · Mar 272012

It is fairly common to represent your vectors as columns then you could represent your "co-vectors" (members of the dual spacethe 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 coursethat isas Office Shredder sayspurely arbitrary- you could always represent the vectors by rowsthe funtionals by columns and do the product the other way around.

AlephZero · Mar 272012

HallsofIvy said:

It is fairly common to represent your vectors as columns then you could represent your "co-vectors" (members of the dual spacethe 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 coursethat isas Office Shredder sayspurely arbitrary- you could always represent the vectors by rowsthe 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 scalarbut
\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 · Mar 282012

Thanks a lot.

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

HallsofIvy · Mar 282012

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

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

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