ECEN601 - Chapter 5 Linear Transformations and Operators...

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Chapter 5 Linear Transformations and Operators 5.1 The Algebra of Linear Transformations Theorem 5.1.1. Let V and W be vector spaces over the feld F .L e t T and U be two linear transFormations From V into W .TheFunc t ion ( T + U ) defned pointwise by ( T + U )( v )= Tv + Uv is a linear transFormation From V into W .±ur thermore ,i F s F ,theFunction ( sT ) defned by ( sT v s ( ) is also a linear transFormation From V into W .These to Fa l ll ineartrans Forma t From V into W , together with the addition and scalar multiplication defnedabove , is a vector space over the feld F . ProoF. Suppose that T and U are linear transformation from V into W .For ( T + U ) de±ned above, we have ( T + U sv + w T ( sv + w )+ U ( sv + w ) = s ( Tw + s ( Uw = s ( + )+( + ) = s ( T + U ) v +( T + U ) w , 80
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5.1. THE ALGEBRA OF LINEAR TRANSFORMATIONS 81 which shows that ( T + U ) is a linear transformation. Similarly, we have ( rT )( sv + w )= r ( T ( sv + w )) = r ( s ( Tv )+( Tw )) = rs ( )+ r ( ) = s ( r ( )) + ( w ) = s (( ) v ) w which shows that ( ) is a linear transformation. To verify that the set of linear transformations from V into W together with the operations de±ned above is a vector space, one must directly check the conditions of De±nition 3.3.1. These are straightforward to verify, andweleaveth isexerc ise to the reader. We denote the space of linear transformations from V into W by L ( V,W ) .No te that L ( ) is de±ned only when V and W are vector spaces over the same ±eld. Fact 5.1.2. Let V be an n -dimensional vector space over the feld F ,andle t W be an m -dimensional vector space over F .T h e n t h e s p a c e L ( ) is fnite- dimensional and has dimension mn . Theorem 5.1.3. Let V , W ,and Z be vector spaces over a feld F .Le t T L ( ) and U L ( W, Z ) h e n t h ec o m p o s e d F u n c t i o n UT defned by ( v U ( T ( v )) is a linear transFormation From V into Z . ProoF. Let v 1 ,v 2 V and s F .Then ,wehave ( sv 1 + v 2 U ( T ( sv 1 + v 2 )) = U ( sTv 1 + 2 ) = sU ( 1 U ( 2 ) = s ( v 1 v 2 ) , as desired. Defnition 5.1.4. IF V is a vector space over the feld F ,a linear operator on V is alineartransFormationFrom V into V .
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82 CHAPTER 5. LINEAR TRANSFORMATIONS AND OPERATORS Defnition 5.1.5. Al ineartrans forma t ion T from V into W is called invertible if there exists a function U from W to V such that UT is the identity function on V and TU is the identity function on W .I f T is invertible, the function U is unique and is denoted by T - 1 .Fur thermore , T is invertible if and only if 1. T is one-to-one: Tv 1 = 2 = v 1 = v 2 2. T is onto: the range of T is W . Example 5.1.6. Consider the vector space V of semi-in±nite real sequences R ω where v =( v 1 ,v 2 3 ,... ) V with v n R for n N .L e t L : V V be the left-shift linear transformation de±ned by Lv v 2 3 4 ) and R : V V be the right-shift linear transformation de±ned by Rv =(0 1 2 ) . Notice that L is onto but not one-to-one and R is one-to-one but not onto. Therefore, neither transformation is invertible.
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This note was uploaded on 09/26/2011 for the course ECEN 601 taught by Professor Staff during the Fall '08 term at Texas A&M.

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ECEN601 - Chapter 5 Linear Transformations and Operators...

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