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# HW 6 - x R v W = p x ∈ P 4 | p(1 = 0(derivative ≤ R x R...

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MATH 225 HOMEWORK 6 pp. 265 – 266 #1, 3, 7, 9, 14, 24, 26, 27 pp. 279 – 281 #7 – 9, 12, 17, 21, 24, 26, 30, 32, 45, 47, 51 A. In V F show that span({ α 1 , . . . , k }) + span({ β 1 , . . . , m }) = span({ 1 , . . . , k , 1 , . . . , m }). B. For 1 , . . . , k V F show that span({ 1 , . . . , k }) is the span of each of the following: i) 1 , 1 + 2 , 2 + 3 , . . . , k -1 + k ; ii) 1 , c 2 1 + 2 , c 3 1 + 3 , . . . , c k 1 + k for any choices of c j F ; iii) 1 , 1 + 2 , 1 + 2 + 3 , . . . , 1 + 2 + + k . C. In each case find two different minimal spanning sets for W V F (observe that each W is a subspace and let P n = { a n x n + + a 1 x + a 0 R [ x ]}): i) W = { A M 3 ( F ) | tr( A ) = A 11 + A 22 + A 33 = 0} M 3 ( F ) F ; ii) W = { A M 4 ( R ) | A T = A } M 4 ( R ) R ; iii) W = { p ( x ) P n | p (1) = 0} R [ x ] R ; iv) W = { p ( x ) P n | p (2) = 0} R
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Unformatted text preview: [ x ] R ; v) W = { p ( x ) ∈ P 4 | p '(1) = 0 (derivative!)} ≤ R [ x ] R ; vi) W = {[ a 1 , . . . , a n ] ∈ R 1x n | a 1 + ⋅ ⋅ ⋅ + a n = 0} ≤ R 1x n R . D. In each case determine if the functions in C ∞ ( R ) are linearly independent: i) sin ( x ), cos (1 + x ), sin (1 + x ); ii) sin ( x ), cos ( x ), sin (2 x ); iii) 1, sin ( x ), xsin ( x ); iv) e 2 x , e –2 x ; e x . Hand In 6A) pp. 265-266 #9; 24; 26; pp. 279 – 281 #12; 24; 26; 30; Bi); Cii); Di)ii). 6B) p.280 #45; Biii); Ciii); Cvi). Due Friday October 7 ....
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