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Notation for vectors 
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#1
Aug2511, 05:40 AM

PF Gold
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What I usually mean by a vector, x, is a quantity which can be written (using Einstein summation convention) as x^{i} e_{i} = x^{i'} e_{i'} and so on. In other words the scalar components {x^{i}} 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 [x_{1}, x_{2}...]^{T}, and to the column vector [x_{1'}, x_{2'}...]^{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.] 


#2
Aug2511, 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]". 


#3
Aug2511, 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 4vectors, when doing a rotation. 


#4
Aug2511, 01:04 PM

PF Gold
P: 936

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 baseindependent 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. 


#5
Aug2511, 02:22 PM

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Thanks
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#6
Aug2611, 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. 


#7
Aug2711, 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 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. 


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