If a line is a linear approximation to a function

Davon Irwin 2022-07-01 Answered
If a line is a linear approximation to a function in 1 variable and a hyperplane is the linear approximation to a function in 2 variables what is the linear approximation to a function in 3 variables?
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Answers (1)

humbast2
Answered 2022-07-02 Author has 21 answers
Your terminology is slightly off.
The linear approximation to a function of 1 variable is indeed a line.
The linear approximation to a function of 2 variables forms a plane.
The linear approximation to a function of any number of variables is a hyperplane.
Unlike the terms "line" and "plane", a hyperplane does not traditionally have a fixed dimension. Instead, the dimensionality depends on context: in R 2 , a hyperplane has one dimension; in R 3 , it has two; and, in general, a hyperplane has one fewer dimension than the ambient space (a nice term for this is "codimension 1").
In general, we don't bother to describe lines as "hyperplanes in 2 D space," mainly because one cannot distinguish between dimension 1 and codimension 1 when the ambient space has 2 dimensions. At the same time, it's not wrong.
The distinction is relevant in 3 D and above, and so we do describe planes as "hyperplanes in 3 D space."
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