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Engineering Calculus Notes 240

Engineering Calculus Notes 240 - −→ x In other words an...

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228 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Thus we see that there are three ways to think of the action of the linear function : R 3 R on a vector −→ x R 3 : Substitute the components of −→ x into a homogeneous polynomial of degree one, whose coefficients are the values of on the standard basis; Multiply the coordinate matrix of −→ x by the matrix representative of ; Take the dot product of the vector −→ a (obtained from the row matrix [ ] by introducing commas) with the vector −→ x . Affine Functions Finally, we introduce one more piece of terminology: an affine function is the sum of a constant and a linear function: φ ( −→ x ) = c +
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Unformatted text preview: ( −→ x ) . In other words, an aFne function is the same thing as a polynomial of degree one (with no homogeneity conditions—that is, without any restriction on the constant term). Note that if φ ( −→ x ) = c + ℓ ( −→ x ) is an aFne function, then for any two vectors −→ x and −→ y , φ ( −→ y ) − φ ( −→ x ) = ℓ ( −→ y ) − ℓ ( −→ x ) = ℓ ( −→ y − −→ x ); setting △ −→ x = −→ y − −→ x , so that −→ y = −→ x + △ −→ x we can write φ ( −→ x + △ −→ x ) = φ ( −→ x ) + ℓ ( △ −→ x ) . (3.1)...
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