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Notation for vectors

by Philip Wood
Tags: notation, vectors
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Philip Wood
#1
Aug25-11, 05:40 AM
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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.]
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CompuChip
#2
Aug25-11, 06:37 AM
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For matrices I've seen notations like
[tex]A_\mathscr{B}[/tex] or [tex][A]_\mathscr{B}[/tex]
for something like "the matrix representation of the linear form A with respect to basis [itex]\mathscr{B}[/itex]".
BruceW
#3
Aug25-11, 08:18 AM
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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
#4
Aug25-11, 01:04 PM
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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.
tiny-tim
#5
Aug25-11, 02:22 PM
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xT and x'T ?

(as at http://en.wikipedia.org/wiki/Transpose)
BruceW
#6
Aug26-11, 04:23 PM
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I was talking about:
[tex] x = x^i \ e_i = {x^i}^\prime \ {e_i}^\prime = x^\prime [/tex]
If you're asking for a notation for just the components of a vector (without the base vectors), then I would just use: [itex]x_i[/itex] or [itex]x_i'[/itex]
The index is left over, like a dummy variable, so it is a notation which refers to any one of the components.
BruceW
#7
Aug27-11, 12:26 PM
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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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