# Notation for vectors

by Philip Wood
Tags: notation, vectors
 P: 850 What I usually mean by a vector, x, is a quantity which can be written (using Einstein summation convention) as xi ei = xi' ei' and so on. In other words the scalar components {xi} change according to the set of base vectors {ei} 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 [x1', x2'...]T. It would be very confusing to use x again as the name for any one 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.]
 Sci Advisor HW Helper P: 4,301 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 $\mathscr{B}$".
 HW Helper P: 3,207 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.
P: 850

## Notation for vectors

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.
 Sci Advisor HW Helper Thanks PF Gold P: 26,113 xT and x'T ? (as at http://en.wikipedia.org/wiki/Transpose)
 HW Helper P: 3,207 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 any one of the components.
 HW Helper P: 3,207 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 ei and ei' 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.

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