Engineering Calculus Notes 450

Engineering Calculus Notes 450 - 438 CHAPTER 4. MAPPINGS...

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Unformatted text preview: 438 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS (c) L is the least number satisfying Equation (4.8). 5. Find the operator norm L for each linear map L below: (a) L(x, y ) = (y, x). (b) L(x, y ) = (x + y, x − y ). √ (c) L(x, y ) = (x + y 2, x). (d) L: R2 → R2 is reflection across the diagonal x = y . (e) L: R2 → R3 defined by L(x, y ) = (x, x − y, x + y ). (f) L: R3 → R3 defined by L(x, y, z ) = (x, x − y, x + y ). 4.3 Linear Systems of Equations A system of equations can be viewed as a single equation involving a mapping. For example, the system of two equations in three unknowns x +2y +5z = 5 2x + y +7z = 4 can be viewed as the vector equation → → L(− ) = − x y where L: R3 → R2 is the mapping given by [L(x, y, z )] = x + 2y + 5z 2x + y + 7z and the right-hand side is the vector with → [− ] = y 5 4 ; we want to solve for the unknown vector with x → [− ] = y . x z We can think of the solution(s) of this system as the “level set” of the → mapping corresponding to the output value − . y (4.9) ...
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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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