Notation for Vectors in Different Bases

[Philip Wood]

What I usually mean by a vector, x, is a quantity which can be written (using Einstein summation convention) as xⁱ e_i = x^i' e_i' and so on. In other words the scalar components {xⁱ} change according to the set of base vectors {e_i} I choose.

But occasionally, in the context of changing bases (e.g. when dealing with rotations on Euclidian space), I want to refer to the column vector [x₁, x₂...]^T, and to the column vector [x_1', x_2'...]^T. It would be very confusing to use x again as the name for anyone of these column vectors.

Is there any agreement as to different notations for a vector and for a column vector which expresses that vector on a particular basis. [I mean compact notations which don't show individual components.]

 

[CompuChip]

For matrices I've seen notations like
$$A_\mathscr{B}$$ or $$[A]_\mathscr{B}$$
for something like "the matrix representation of the linear form A with respect to basis $\mathscr{B}$".

 

[BruceW]

uh.. why don't you use x for one vector and x' for the rotated vector?
I think they often use this in the notation of 4-vectors, when doing a rotation.

 

[Philip Wood]

CompuChip Thank you. I'd not seen this.

BruceW Thanks, but the transforms I'm concerned with are passive ones: the same vector expressed on different bases. If I use x and x'to distinguish the column vectors which give the components of the vectors on the two bases, what would I then use for the base-independent vector (what I called x in my original post)? That's what I'm concerned about, notation which distinguishes these two different types of vector, not notation which distinguishes one column vector of components from a column vector of components on a different basis.

 

[tiny-tim]

x^T and x'^T ?

(as at http://en.wikipedia.org/wiki/Transpose" )

 

[BruceW]

I was talking about:
$$x = x^i \ e_i = {x^i}^\prime \ {e_i}^\prime = x^\prime$$
If you're asking for a notation for just the components of a vector (without the base vectors), then I would just use: $x_i$ or $x_i'$
The index is left over, like a dummy variable, so it is a notation which refers to anyone of the components.

 

[BruceW]

I had a look in my textbook, and it says this:

"Thus, we use x and x' to denote different column matrices which, in different bases e_i and e_i' represent the same vector x. In many texts, however, this distinction is not made and x (rather than x) is equated to the corresponding column matrix ; if we regard x as the geometrical entity, however, this can be misleading and so we explicitly make the distinction."

So I guess in my textbook, they use bold for the actual vector, and x or x' to mean the components of the vector in a particular basis.