- #1
Vectors - write ordered triples in vertical or horizontal form?
- Context: High School
- Thread starter Outrageous
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- Form Horizontal Vectors Vertical
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Discussion Overview
The discussion revolves around the representation of vectors in different forms, specifically ordered triples in vertical and horizontal formats. Participants explore the implications of these representations in the context of vector spaces and their properties.
Discussion Character
- Conceptual clarification
- Debate/contested
- Technical explanation
Main Points Raised
- One participant questions whether writing vectors in vertical form (column matrices) versus horizontal form (row matrices) makes a difference.
- Another participant suggests that both forms are valid representations of vectors and mentions the concepts of covariant and contravariant vectors, although they express uncertainty about which is which.
- A more advanced perspective is introduced regarding the dual space of a vector space, explaining how functions from the vector space to a field can be represented as row matrices, while vectors themselves can be represented as column matrices.
- There is a suggestion that for most practical purposes, writing vectors horizontally may be more convenient.
Areas of Agreement / Disagreement
Participants express differing views on the significance of the representation forms, with some suggesting that it may not matter for most applications, while others introduce more complex theoretical considerations. No consensus is reached on the implications of these representations.
Contextual Notes
The discussion touches on advanced concepts such as dual spaces and the isomorphism between vector spaces and their duals, which may not be fully understood by all participants. There is also a mention of uncertainty regarding the definitions of covariant and contravariant vectors.
Who May Find This Useful
This discussion may be of interest to students and professionals in mathematics and physics, particularly those exploring vector spaces and their representations.
- #2
tiny-tim
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Hi Outrageous! 
(i'm puzzled as to why you said "complex" … you're not thinking of quaternions, are you?
)
horizontal or vertical are both vectors
one is covariant, the other is contravariant (i can never remember which is which
)
(i'm puzzled as to why you said "complex" … you're not thinking of quaternions, are you?
)Outrageous said:
horizontal or vertical are both vectors
one is covariant, the other is contravariant (i can never remember which is which
)for most purposes, it doesn't matter, so you might as well write everything horizontally, since that's more convenient! 
- #3
Outrageous
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Thank you. That should be vector.
How to edit title?
How to edit title?
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- #4
tiny-tim
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i don't think can edit the title
(but you can edit the first post, to say "ignore the title!")
(but you can edit the first post, to say "ignore the title!"
)- #5
Mark44
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Title changed
- #6
HallsofIvy
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This is a bit more abstract and advanced but one way of looking at it is this: Given an n-dimensional vector space, V, over field F, the set of all functions from V to F, the "dual space" to V, is itself a vector space, V*, with addition defined by (f+ g)(v)= f(v)+ g(v) and (af)v= a(f(v)), also of dimenion n. We can then represent functions in V* as "row matrices" and the vectors in V as "column matrices" so that the operation f(v) is a matrix multipication.
However, because it is still true that V and V*, both being n-dimensional vector spaces, are isomorphic we can identify one with the other, the row and column matrices as both representing vectors and think of the matrix multiplication as an "inner product" on a vector space.
However, because it is still true that V and V*, both being n-dimensional vector spaces, are isomorphic we can identify one with the other, the row and column matrices as both representing vectors and think of the matrix multiplication as an "inner product" on a vector space.
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- #7
Outrageous
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Really advanced. Thank you.
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