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Some notation I want explained

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It''s irritating when I see some math notation that I don''t understand, so, ya know what, I''ll come back to this very thread whenever I have a question about that. Confusing notation is irritating, and having made the _HUGE_ progress that I have in math, I must have it out of the way. ...THE COLON... In 3D GAME ENGINE DESIGN (Eberly, 2000), Dave Eberly uses this notation: M : R^3 --> R^3 Here, we have a matrix M, and R^3 represents the set of 3-coordinate vectors. Simple enough. In this case, M represents a linear transformation. But what does that colon really mean? I''ve seen this symbol used in function definitions as well, and it''s somewhat irritating when they use '':'' instead of ''=''. Are they the same? similar?

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I''ve seen it used plenty without really knowing its exact meaning but I guess it''s just notation for something like "is a X" or "is of type X." It doesn''t make sense to me to use = in its place here, I mean just read it. "M equals a linear transformation in R3" ???

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R is the set of real numbers. R^2 is two-dimensional space or the Cartesian coordinate plane. R^3 is familiar three-space. Generally speaking, R^N is n-space. This notation says that the transformation M transforms coordinate triples from three-space to three-space.

EDIT: Spacing Issues

[edited by - TThirion on July 24, 2002 8:28:26 PM]

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The colon is interpreted as meaning such that . So, in functional notation, the statement f:Rm- > Rn is interpreted as saying, "A function f, such that f maps the space Rm onto the space Rn".

Cheers,

Timkin

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I like to think of this as a sort of function prototype for math. It tells you what a function f expects in terms of number of arguments, and tells you how many things it outputs.

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It means "M is a function that maps from R^3 to R^3". See this page at Mathworld for a detailed description of this notation and what it means.

It also means "such that'' in other contexts, e.g.

R^+ = {x : X in R, x > 0}

says "R^+ is the set of all x such that x is a real number and x is positive", i.e. the set of positive reals. This also uses a colon but the layout of the expression is quite different.

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