2.22.3 - Homework 4 - Solutions 2.2 Problem 1 1. TRUE. See...

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Homework 4 - Solutions 2.2 Problem 1 1. TRUE. See Theorem 2.7 (a). 2. TRUE. See last paragraph on page 80. 3. FALSE. It will be a n × m matrix. 4. TRUE. See Theorem 2.8 (a). 5. TRUE. See Theorem 2.7. 6. FALSE. L ( V,W ) is a collection of functions from V to W ,and L ( W, V ) is a collection of functions from W to V . These are not the same unless V and W are the same. 2.2 Problem 4. T ± 10 00 ² =1 T ± 01 ² =1+ x 2 T ± ² =0 T ± ² =2 x The coordinates of these four polynomials with respect to the basis β form the four columns of the matrix [ T ] γ β : [ T ] γ β = 1100 0002 0100 2.2 Problem 9. Let z 1 = a 1 + ib 1 , z 2 = a 2 + ib 2 α R ..Th en T ( z 1 + αz 2 )= T (( a 1 + ib 1 )+ α ( a 2 + ib 2 )) = T (( a 1 + αa 2 i ( b 1 + αb 2 )) =( a 1 + αa 2 ) i ( b 1 + αb 2 ) a 1 ib 1 α ( a 2 ib 2 ) = z 1 + α z 2 = T ( z 1 αT ( z 2 ) This proves T is linear. T (1) = 1, and T ( i i . In the basis { 1 ,i } , the coordinates for T (1) and T ( i )are(1 , 0) and (0 , 1) (written as columns). Therefore, [ T ] β = ± 0 1 ²
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2.2 Problem 10. The j th column of [ T ] β is obtained by expressing T ( v j ) in the basis β .S in c e T ( v j )= v j + v j - 1 ,weget T ( v j )=0 v 1 + ··· +1 v j - 1 v j 0 v n The coordinates for T ( v j ) in the basis β are (0 ,..., 1 , 1 0), where the only nonzero entries are at the j 1and j positions (written as a column vector). For the special case j = 1, the only nonzero entry is the ±rst entry (since v j - 1 is taken to be 0). Therefore, [ T ] β = 1100 0 0110 0 0011 0 . . .0 0 011 0 01 This is a matrix which has (a) 1 on each diaginal entry and each entry above a diagonal entry (b) zero in all other entries. 2.2 Problem 16. Let the dimension of V and W be n . Assume that the rank of T is k .L e t S = { v 1 ,...,v n - k } be a basis of N ( T ). Extend S to a basis β = { v 1 n - k ,v n - k +1 n } of V (by Corollary on Page 51). De±ne w j = T ( v j )for j = n k ,...,n .Le t R = { w n - k +1 ,...,w n } .
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This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.

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2.22.3 - Homework 4 - Solutions 2.2 Problem 1 1. TRUE. See...

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