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Engineering Calculus Notes 238

# Engineering Calculus Notes 238 - in a standard way a linear...

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226 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION These two properties together can be summarized by saying that h ( −→ x ) respects linear combinations : for any two vectors −→ x and −→ y and any two numbers α and β , h ( α −→ x + β −→ y ) = αh ( −→ x ) + βh ( −→ y ) . A function which respects linear combinations is called a linear function . The preceding discussion shows that every homogeneous polynomial of degree one is a linear function. Recall that the standard basis for R 3 is the collection −→ ı, −→ , −→ k of unit vectors along the three positive coordinate axes; we will find it useful to replace the “alphabetical” notation for the standard basis with an indexed one: −→ e 1 = −→ ı −→ e 2 = −→ −→ e 3 = −→ k. The basic property 4 of the standard basis is that every vector −→ x R 3 is, in a standard way, a linear combination of these specific vectors:
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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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