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Week4-7 - Chapter 2 Matrices and Determinants 1 Linear...

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Chapter 2: Matrices and Determinants March 16, 2009 1 Linear transformations Lecture 7 Definition 1.1. Let X and Y be nonempty sets. A function from X to Y is a rule f : X Y such that each element x in X is assigned a unique element y in Y , written as y = f ( x ); the element y is called the image of x under f , and the element x is called the preimage of f ( x ). Functions are also called maps , or mappings , or transformations . Definition 1.2. A function T : R n R m is called a linear transformation if for any vectors u , v in R n and scalar c , (a) T ( u + v ) = T ( u ) + T ( v ), (b) T ( c u ) = cT ( u ). Example 1.1. (a) The function T : R 2 R 2 , defined by T ( x 1 , x 2 ) = ( x 1 + 2 x 2 , x 2 ), is a linear transformation, see Figure 1 (1,0) x x y y (0,0) (0,1) (1,1) (0,0) (1,0) (2,1) (3,1) Figure 1: The geometric shape under a linear transformation. (b) The function T : R 2 R 3 , defined by T ( x 1 , x 2 ) = ( x 1 + 2 x 2 , 3 x 1 + 4 x 2 , 2 x 1 + 3 x 2 ), is a linear transformation. (c) The function T : R 3 R 2 , defined by T ( x 1 , x 2 , x 3 ) = ( x 1 + 2 x 2 + 3 x 3 , 3 x 1 + 2 x 2 + x 3 ), is a linear transformation. Example 1.2. The transformation T : R n R m by T ( x ) = A x , where A is an m × n matrix, is a linear transformation. Example 1.3. The map T : R n R n , defined by T ( x ) = λ x , where λ is a constant, is a linear transformation, and is called the dilation by λ . 1
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Example 1.4. The refection T : R 2 R 2 about a straight line through the origin is a linear transformation. Example 1.5. The rotation T : R 2 R 2 about an angle θ is a linear transformation, see Figure 3. θ O θ θ y x u T(u+v) T(u) T(v) u+v v Figure 2: Rotation about an angle θ . Proposition 1.3. Let T : R n R m be a linear transformation. Then for any vectors v 1 , v 2 , . . . , v k in R n and scalars c 1 , c 2 , . . . , c k , we have T ( c 1 v 1 + v 2 + · · · + c k v k ) = c 1 T ( v 1 ) + c 2 T ( v 2 ) + · · · + c k T ( v k ) . In particular, T ( 0 ) = 0 . Theorem 1.4. Let T : R n R m be a linear transformation. Given vectors v 1 , v 2 , . . . , v k in R n . (a) If v 1 , v 2 , . . . , v k are linearly dependent, then T ( v 1 ) , T ( v 2 ) , . . . , T ( v k ) are linearly dependent. (b) If T ( v 1 ) , T ( v 2 ) , . . . , T ( v k ) are linearly independent, then v 1 , v 2 , . . . , v k are linearly indepen- dent. 2 Standard matrix of linear transformation Let T : R n R m be a linear transformation. Consider the following vectors e 1 = 1 0 . . . 0 , e 2 = 0 1 . . . 0 , . . . , e n = 0 0 . . . 1 in the coordinate axis of R n . It is clear that for every vector x = x 1 x 2 . . . x n = [ x 1 , x 2 , . . . , x n ] T is a linear combination of e 1 , e 2 , . . . , e n , i.e., x = x 1 e 1 + x 2 e 2 + · · · + x n e n . 2
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Then it follows from linearity that T ( x ) = x 1 T ( e 1 ) + x 2 T ( e 2 ) + · · · + x n T ( e n ) . This means that T ( x ) is completely determined by the images T ( e 1 ), T ( e 2 ), . . . , T ( e n ). The ordered set { e 1 , e 2 , . . . , e n } is called the standard basis of R n . Definition 2.1. The standard matrix of a linear transformation T : R n R m is the m × n matrix A = £ T ( e 1 ) , T ( e 2 ) , . . . , T ( e n ) = £ T e 1 , T e 2 , . . . , T e n .
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