Engineering Calculus Notes 433

Engineering Calculus Notes 433 - 421 4.1. LINEAR MAPPINGS...

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Unformatted text preview: 421 4.1. LINEAR MAPPINGS The respective matrix representatives are 110 011 L′ = 1 −1 [ L] = 1 1 2 −1 , so 1 −1 [L] · L′ = 1 1 · 2 −1 110 011 1 0 −1 = 1 2 1 2 2 −1 and L′ · [ L] = 110 011 1 −1 · 1 1 = 2 −1 20 32 . You should verify that these last two matrices are, in fact the matrix representatives of L ◦ L′ and L′ ◦ L, respectively. Exercises for § 4.1 Practice problems: 1. Which of the following maps are linear? Give the matrix representative for those which are linear. (a) f (x, y ) = (y, x) (b) f (x, y ) = (x, x) (c) f (x, y ) = (ex cos y, ex sin y ) (d) f (x, y ) = (x2 + y 2 , 2xy ) (e) f (x, y ) = (x + y, x − y ) (f) (i) f (x, y ) = (x, y, x2 − y 2 ) (j) (g) f (x, y ) = (x + y, 2x − y, x + 3y ) (k) f (x, y, z ) = (2x + 3y + 4z, x + z, y + z ) (m) f (x, y, z ) = (x − 2y + 1, y − z + 2, x − y − z ) f (x, y ) = (x, y, x2 − y 2 ) (h) f (x, y ) = (x − 2y, x + y − 1, 3x + 5y ) (l) f (x, y ) = (x, y, xy ) f (x, y, z ) = (y + z, x + z, x + y ) (n) f (x, y, z ) = (x + 2y, z − y + 1, x) ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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