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Unformatted text preview: 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 . Let T and U be two linear transFormations From V into W . The Function ( T + U ) defned pointwise by ( T + U ) ( v ) = Tv + Uv is a linear transFormation From V into W . urthermore, iF s F , the Function ( sT ) defned by ( sT ) ( v ) = s ( Tv ) is also a linear transFormation From V into W . The set oF all linear transFormation From V into W , together with the addition and scalar multiplication defned above, is a vector space over the feld F . ProoF. Suppose that T and U are linear transformation from V into W . For ( T + U ) dened above, we have ( T + U ) ( sv + w ) = T ( sv + w ) + U ( sv + w ) = s ( Tv ) + Tw + s ( Uv ) + Uw = s ( Tv + Uv ) + ( Tw + Uw ) = s ( T + U ) v + ( T + U ) w , 80 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 ( Tv ) + r ( Tw ) = s ( r ( Tv )) + rT ( w ) = s (( rT ) v ) + ( rT ) w which shows that ( rT ) is a linear transformation. To verify that the set of linear transformations from V into W together with the operations dened above is a vector space, one must directly check the conditions of Denition 3.3.1. These are straightforward to verify, and we leave this exercise to the reader. We denote the space of linear transformations from V into W by L ( V,W ) . Note that L ( V,W ) is dened only when V and W are vector spaces over the same eld. Fact 5.1.2. Let V be an ndimensional vector space over the feld F , and let W be an mdimensional vector space over F . Then the space L ( V,W ) is fnite dimensional and has dimension mn . Theorem 5.1.3. Let V , W , and Z be vector spaces over a feld F . Let T L ( V,W ) and U L ( W,Z ) . Then the composed Function UT defned by ( UT ) ( v ) = U ( T ( v )) is a linear transFormation From V into Z . ProoF. Let v 1 ,v 2 V and s F . Then, we have ( UT ) ( sv 1 + v 2 ) = U ( T ( sv 1 + v 2 )) = U ( sTv 1 + Tv 2 ) = sU ( Tv 1 ) + U ( Tv 2 ) = s ( UT ) ( v 1 ) + ( UT ) ( v 2 ) , as desired. Defnition 5.1.4. IF V is a vector space over the feld F , a linear operator on V is a linear transFormation From V into V . 82 CHAPTER 5. LINEAR TRANSFORMATIONS AND OPERATORS Defnition 5.1.5. A linear transformation 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 . If T is invertible, the function U is unique and is denoted by T 1 . Furthermore, T is invertible if and only if 1. T is onetoone: Tv 1 = Tv 2 = v 1 = v 2 2. T is onto: the range of T is W ....
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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.
 Fall '08
 Staff

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