MAT1341-L18-LinearTransformations0

MAT1341-L18-LinearTransformations0 - LINEAR TRANSFORMATIONS...

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Unformatted text preview: LINEAR TRANSFORMATIONS JOS ´ E MALAG ´ ON-L ´ OPEZ In this section we deal with the adequate notion of a function between vector spaces, where by adequate we mean the functions that preserve linear combinations, and hence the structure of a vector space. Functions Recall that a function f : D → R from a set D to a set R is a rule that associates to every element in D a unique element in R . • The set D is called the domain of f , and its elements are called the input values . • The set R is called the range of f . • The output value f ( x ) corresponding to the input x under f is called the image of x . In such case, we also say that f maps x into f ( x ). Given a function f : D → R the set f ( D ) = { y ∈ R | y = f ( x ) , for some x in D } is called the image of D under f . Example 1. Some examples of functions are: • f : R → R given by f ( x ) = cos( x ). • f : R 2 → R given by f ( x, y ) = x + y . • f : R 2 → R 2 given by f ( x, y ) = (2 x, x + y ). • f : R → R 2 given by f ( x ) = (2 x, − x ). 1 Example 2. Some more elaborated examples of functions are: • Fix a positive integer n . Then f : M n × n → M n × n , given by f ( A ) = det( A ) defines function. • Given a fixed vector vectorv . Then T : R n → R , given by T ( vectorx ) = vectorx • vectorv defines a function....
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This note was uploaded on 01/22/2011 for the course MAT 1341 taught by Professor Josemalagonlopez during the Fall '10 term at University of Ottawa.

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MAT1341-L18-LinearTransformations0 - LINEAR TRANSFORMATIONS...

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