Lecture4

# Lecture4 - 20 Lecture 4 Linear Maps We have all seen linear...

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Lecture 4 Linear Maps We have all seen linear maps before. In fact, most of the maps we have been using in Calculus are linear. 4.1 Two Important Examples 4.1.1 The Integral To integrate the function f ( x ) = x 2 + 3 x - cos x over the interval [ a,b ], we ﬁrst ﬁnd the antiderivative of x 2 , that is 1 3 x 3 , then the antiderivative of x , which is 1 2 x 2 , and then multiply that by 3 to get 3 2 x 2 . Finally, we ﬁnd the antiderivative of cos x , which is sin x , and then multiply that by - 1 to get - sin x . To ﬁnish the problem we insert the endpoints. Thus, Z 1 - 1 x 2 + 3 x - cos xdx = Z 1 - 1 x 2 dx + 3 Z 1 - 1 xdx - Z 1 - 1 cos xdx = 1 3 x 3 1 - 1 + 3 2 x 2 1 - 1 - [sin x ] 1 - 1 = 2 3 - sin1 + sin( - 1) . What we have used is the fact that the integral is a linear map C ([ a,b ]) -→ R and that 21
22 LECTURE 4. LINEAR MAPS Z b a rf ( x ) + sg ( x ) dx = r Z b a f ( x ) dx + s Z b a g ( x ) dx. 4.1.2 The Derivative Another example is diﬀerentiation Df = f 0 . To diﬀerentiate the function f ( x ) = x 4 - 3 x + e x - cos x , we ﬁrst diﬀerentiate each term of the function and then add: D ( x 4 - 3 x + e x - cos x ) = Dx 4 - 3 Dx + De x - D cos x = 4 x 3 - 3 + e x + sin x. Deﬁnition. Let V and W be two vector spaces. A map T : V -→ W is said to be linear if for all v,u V and all r,s R we have: T ( rv + su ) = rT ( v ) + sT ( u ) . Remark: This can also be written by using two equations:

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## This note was uploaded on 11/23/2011 for the course MATH 2025 taught by Professor Staff during the Spring '08 term at LSU.

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Lecture4 - 20 Lecture 4 Linear Maps We have all seen linear...

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