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linear algebra question I know what R^3 or R^2 or R is but what is F, F^infinite and F1, F2, F3 ?? stuff like that i dont understand

Mathematics
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F is usually a field, R^n being an example and other well known being complex or rational numbers .
when you say its a field what does that mean? i gotta see like an example for it.. For example i know that R^2 = (a,b) <-- dimension of two but i dont know how to do F stuff is there some kind of chart?
F is just an arbitrary field. When you say\[\mathbb{R}^2\]you mean\[(a,b), a,b\in \mathbb{R}\]all ordered pairs where a and b are in R. When you say\[\mathbb{F}^2\]we mean\[(a,b),a,b\in\mathbb{F}\]all ordered pairs where a,b come from the same field.

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oh i see but my teacher do like some insane stuff like \[F2= sinx, e^x \]
what would \[F^\infty \] would it be?
\[\mathbb{F}^\infty \]would be the set of all infinite sequences. You could write it down as:\[(a_1,a_2,a_3,a_4,\ldots\]where\[a_i\in\mathbb{F}\]
oh so \[F\] is just another arbituary field like \[R \]
Right. In most of linear algebra, we never use the properties that are specific to R (things like completeness, etc). We only use the field properties of R. So in the theory of linear algebra, we can talk in the most general of settings, an arbitrary field.
Everything we prove will of course work if we are in R, but it could also work for any field, whether its the complex numbers, or even finite fields.
i get it know but is there like a structure i need to follow? for example \[P2 =[ax^2+bx+c | a,b,c \in R ]\]
and like M2X2 is using the matrices.... is there any particular for \[F\]

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