MA353_HW13_sol

MA353_HW13_sol - Purdue University MA 353 Linear Algebra II...

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Unformatted text preview: Purdue University MA 353: Linear Algebra II with Applications Homework 13, due Apr. 23, Some solutions Sec. 6.3:#2(c): Let V = P 2 ( R ) and define an inner product on V by h f,h i = Z 1 f ( t ) h ( t ) dt. Define a linear transformation g : V → R by g ( f ) = f (0)+ f (1). Find a vector h ∈ V such that g ( f ) = h f,h i for all f ∈ V . Answer. Recall, from the proof of Theorem 6.8, that the vector h can be constructed as follows: Let { h 1 ( t ) ,h 2 ( t ) ,h 3 ( t ) } be an orthonormal basis of V . Then h ( t ) = g ( h 1 ) h 1 ( t ) + g ( h 2 ) h 2 ( t ) + g ( h 3 ) h 3 ( t ) = ( h 1 (0) + h 1 (0)) h 1 ( t ) + ( h 2 (0) + h 2 (0)) h 2 ( t ) + ( h 3 (0) + h 3 (0)) h 3 ( t ) . Hence we need to construct an orthonormal basis { h 1 ,h 2 ,h 3 } . For this, apply the Gram-Schmidt process to the standard basis { 1 ,t,t 2 } of V . I omit the detail. #3(c): Let V = P 1 ( R ) and define an inner product on V by h f,g i = Z 1- 1 f ( t ) g ( t ) dt. Define a linear operator T : V → V by T ( f ) = f + 3 f . Then compute T * (4- 2 t ). Answer. Let β = { 1 √ 2 , q 3 2 t } . Then one can see that β is an orthonor- mal basis of V with respect to this inner product. Then [ T ] β = [ T ( 1 √ 2 )] β [ T ( q 2 3 t )]...
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MA353_HW13_sol - Purdue University MA 353 Linear Algebra II...

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