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ECEN601-Chapter5 - Chapter 5 Linear Transformations and...

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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 field F . Let T and U be two linear transformations from V into W . The function ( T + U ) defined pointwise by ( T + U ) ( v ) = Tv + Uv is a linear transformation from V into W . Furthermore, if s F , the function ( sT ) defined 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 defined above, is a vector space over the field F . Proof. Suppose that T and U are linear transformation from V into W . For ( T + U ) defined 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
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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 ( 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 defined above is a vector space, one must directly check the conditions of Definition 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 defined only when V and W are vector spaces over the same field. Fact 5.1.2. Let V be an n -dimensional vector space over the field F , and let W be an m -dimensional vector space over F . Then the space L ( V, W ) is finite- dimensional and has dimension mn . Theorem 5.1.3. Let V , W , and Z be vector spaces over a field F . Let T L ( V, W ) and U L ( W, Z ) . Then the composed function UT defined 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. Definition 5.1.4. If V is a vector space over the field F , a linear operator on V is a linear transformation from V into V .
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82 CHAPTER 5. LINEAR TRANSFORMATIONS AND OPERATORS Definition 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 one-to-one: Tv 1 = Tv 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-infinite real sequences R ω where v = ( v 1 , v 2 , v 3 , . . . ) V with v n R for n N . Let L : V V be the left-shift linear transformation defined by Lv = ( v 2 , v 3 , v 4 , . . . ) and R : V V be the right-shift linear transformation defined by Rv = (0 , v 1 , v 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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