Unformatted text preview: in a standard way, a linear combination of these speciFc vectors: −→ x = ( x,y,z ) = x −→ ı + y −→ + z −→ k or ( x 1 ,x 2 ,x 3 ) = x 1 −→ e 1 + x 2 −→ e 2 + x 3 −→ e 3 . Then combining this with the fact that linear functions respect linear combinations, we easily see (Exercise 8 ) that all linear functions are homogeneous polynomials in the coordinates of their input: Remark 3.2.1. Every linear function ℓ : R 3 → R is determined by its eFect on the elements of the standard basis for R 3 : if ℓ ( −→ e i ) = a i , for i = 1 , 2 , 3 then ℓ ( x 1 ,x 2 ,x 3 ) is the degree one homogeneous polynomial ℓ ( x 1 ,x 2 ,x 3 ) = a 1 x 1 + a 2 x 2 + a 3 x 3 . 4 No pun intended...
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 Fall '08
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 Calculus, Linear Algebra, Vectors, Linear function, Standard basis

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