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# sol - SOLUTIONS FOR HOMEWORK 1 Section 1.1 1(a The...

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SOLUTIONS FOR HOMEWORK 1 Section 1.1 1. (a) The vectors (3 , 1 , 2) and (6 , 4 , 2) are parallel iff (this is an abbreviation for “if and only if”) there exists t R s.t. t (3 , 1 , 2) = (6 , 4 , 2). That is, for some t , three equalities must be satisfied: (i) 3 t = 6, (ii) t = 4, and (iii) 2 t = 2. But there is no such t . Thus, the vectors are not parallel . (c) The vectors (5 , 6 , 7) and ( 5 , 6 , 7) are parallel , since the equal- ity t (5 , 6 , 7) = ( 5 , 6 , 7) is satisfied for t = 1. 2. (c) The line in question is the set of all points of the form (3 , 7 , 2)+ t ( (3 , 7 , 8) (3 , 7 , 2) ) = (3 , 7 , 10 t ), with t R . 3. (a) The plane in question is the set of all points of the form (2 , 5 , 1) + t ( (0 , 4 , 6) (2 , 5 , 1) ) + s ( ( 3 , 7 , 1) (2 , 5 , 1) ) = (2 2 t 5 s, 5 + t 2 s, 1 + 7 t + 2 s ) , with t,s R . Section 1.2 1. (h) No . Consider, for instance p ( x ) = 13 + 2 x , and q ( x ) = 2 x . Then deg( p ) = deg( q ) = 1. However, p ( x ) + q ( x ) = 13, hence deg( p + q ) = 0 < 1. 4. (a) parenleftbigg 2 5 3 1 0 7 parenrightbigg + parenleftbigg 4 2 5 5 3 2 parenrightbigg = parenleftbigg 6 3 2 4 3 9 parenrightbigg . (e) (2 x 4 7 x 3 +4 x +3)+(8 x 3 +2 x 2 6 x +7) = 2 x 4 + x 3 +2 x 2 2 x +10. 7. We have to show that f ( t ) = g ( t ) for any t S = { 0 , 1 } . That is, we need to show the equalities f (0) = g (0), and f (1) = g (1). But these are easily verified. The equality f + g = h is handled in the same way.

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sol - SOLUTIONS FOR HOMEWORK 1 Section 1.1 1(a The...

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