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# is the zero function f ( 20 ) But the orthogonal complem and so ( ( P (0 ) ) ) = C / a , 6 ) * P ( ) plement of fo ist andbegan The difference is...

Need to find basis for orthagonal complement of th a following sub spaces(4.4.13)

is the zero function f ( 20 )But the orthogonal complemand so ( ( P (0 ) ) ) = C / a , 6 ) * P ( )plement of fo istandbeganThe difference is that in infinite- dimensional function( of in the eatingcan be better hereas it finite dimensions every prove Will proper lywho led Are )( 2 ) Write downbecubies ous an infinitesimal fraction of the entire vector space is a fourthparadox is interestingly , the reason behind the success of numeric Howard !to the orthogonalsuch as the finite element method , ( 81FOXin function space120 Let WC Vthe Let beanProve thatExercises272 Prove that isthen wit WitNote : In Exercises 4 4 . 12 - 15 , use the dot product .123 1 3 ) Show the4 . 4 . 12 . Find the orthogonal complement WV of the subsindicated vectors . What is the dimension of W in each case ?subspaces VCR spanned by thelines in !( ( ) ( ) ( ) .co ( ) ( ) . ( ) ( ) ( 3 6 /1424 Prove that( See Exercise 2Oulas Fil in the4 .4.13 . Find a basisfor the orthogonal complement of the following subspaces of R ?4126 . Proveplane 3 2 + 4y - 52 - 0 ; ( b ) the line in the direction (2 1 3 ) ; ( C ) the image of thePART Let WC2 0 2 1( d ) the cokernel of the same matrixmenformas an ortheV . then it's art4 . 4 . 14 . Find a basis for the orthogonal complenplement of the following subspaces of R 4 : ( al thand = Cont 1set of solutions to - * + 34 - 2 2 + w( b ) the subspace spanned by ( 1 2 , - 1.31coimage of the same matrixIF , ( - 1 2 0 1 ) ; ( C ) the kernel of the matrix in Exercise 4 . 4 130 ; ( d ) the14.28 . Considerusual It inne4 . 4 . 15 . Decompose each of the following vectors with respect to the indicated subspace as( 6 ) Prove thedimensionalw + 2 , where WEW Z EW ( a ) v( 2 ) . W = span ( ( 3 )( b )( 3 ) . W = span ( ( ) . ( ) : ( ) v = ( ) . W = Ker ( 2 8Orthogonali( 2 ) v = ( 6 ) W = ing ( 2 - 1 6 ) ; ( 6 ) v = ( ) . W = Ker1 0 01 Chapter 2matrix A Ac4 .4 . 16 . Redo Exercise 4 .4 12 using the weighted inner product ( V W ) = 1 20 + 24 2 42 + 3139 ) The second tospace ) and theinstead of the dot product4 . 4 . 17 . Redo Example 4 43 with the dot product replaced by the weighted inner product(8 9 also of( V , W ) = 0 1 20 1 + 2 0 2 2 2 + 3 0 3 2 3 + 4 04 2 4subspaces are$ 4 .4 . 18 . Prove that the orthogothozonal complement Wit of a subspace WV CV is itseTheorem 4B itself a subspacecomplementsNeuronalIn general , a subset W CV of a normed vector space is dense if for every V E V , and even6 3 0 . one can find WE W with ly - will Ce. The Weierstrate Approximation TheoremTheorem 10 . 2 21 , tell us that the polynomials form a dense stopat the space of continentem , 1 19functions , and underlies the proof of the result mentioned in the preceding paragraphProof : A vematrix mal