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Lecture15 - k ∈ R then T r 1 v 1 r 2 v 2 ·· r k v k =...

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University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Winter 2008 Lecture 15 and 16 Section 2.3 Sophie Chrysostomou 1 Linear Transformations definition 1.1 A function T : R n -→ R m is a linear transformation if for all v , u R n and for all r R , the following are satisfied: 1. T ( u + v ) = T ( u ) + T ( v ) 2. T ( r v ) = rT ( u ) If T : R n -→ R m is a linear transformation.Then: 1. R n is the domain of T . 2. R m is the codomain of T . 3. If W R n then: the image of W under T is T [ W ] = { T ( w ) w W } . 4. the range of T is T [ R n ] = { T ( v ) v R n } . 5. If W R m , then the inverse image of W under T is T - 1 [ W ] = { v R n T ( v ) W } The set T - 1 [ { 0 } ] = { v R n T ( v ) = ) } ( where 0 R m ) is called the kernel of T . 1
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example 1.2 Let T : R 3 -→ R 2 be given by T ([ x 1 , x 2 , x 3 ]) = [ x 1 - x 3 , x 2 + x 3 ] . 1. Show that T is a linear transformation. 2. If H = { [ x, x, x ] x R } , find the image of H under T . 3. If U = { [1 , 2] , [ - 1 , 3] } , find the inverse image of U under T . 4. Find the kernel of T . 2
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theorem 1.3 Let T : R n -→ R m be a linear transformation, then: i) if v 1 , v 2 , v 3 , · · · , v k R n and r 1 , r 2 , · · ·
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Unformatted text preview: k ∈ R then T ( r 1 v 1 + r 2 v 2 + ··· + r k v k ) = r 1 T ( v 1 ) + r 2 T ( v 2 ) + ··· + r k T ( v k ) . ii) T ( ) = where ∈ R n and ∈ R m . 4 theorem 1.4 If T : R n-→ R m is a linear transformation, and B = { b 1 , b 2 , ··· , b n } a basis for R n . Then: if v ∈ R n = ⇒ T ( v ) is determined by T ( b 1 ) , T ( b 2 ) , ··· , T ( b n ) . example 1.5 Suppose T : R 2-→ R 2 is a linear transformation and T ([1 , 1]) = [3 , 2] , T ([2 , 3]) = [7 , 7] . Find T ([ x, y ]) . 5 Homework: Suppose T : R 2-→ R 3 is a linear transformation and T ([1 , 2 , 1]) = [1 , 3 , 2] , T ([2 , 3 , 1]) = [2 , 2 , 7] , T ([1 , 1 , 2]). Find T ([ x, y, z ]). 6...
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