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Unformatted text preview: Practice Prelim 2 Solutions 1. Note that P 4 is a 5dimensional linear space with basis B = { 1 ,x,x 2 ,x 3 ,x 4 } . (a) In terms of the basis B , T a b c = a 2 b + 3 c 3 a + 2 b + c a + 2 b c a + c B = 1 2 3 3 2 1 1 2 1 1 1 a b c To find the image (and kernel) of T , rowreduce the matrix of T : 1 2 3 3 2 1 1 2 1 1 1  3 R 1 R 1 R 1 1 2 3 8 8 4 4 2 2 8 4 2 1 2 3 1 1 1 1 1 1  R 2 R 2 1 2 3 1 1 The pivots in the first and second columns indicate these columns are independent in the original matrix, so im( T ) = span 1 3 1 1 B ,  2 2 2 B = span 1 + 3 x + x 2 + x 4 , 2 + 2 x + 2 x 2 A basis for im( T ) is given by 1 + 3 x + x 2 + x 4 , 2 + 2 x + 2 x 2 . (b) Set p 1 ( x ) = 1 + 3 x + x 2 + x 4 and p 2 ( x ) = 2 + 2 x + 2 x 2 . Then p ( x ) = ( a + c ) p 1 ( x ) + ( b c ) p 2 ( x ). So the coordinates with respect to the basis 1 + 3 x + x 2 + x 4 , 2 + 2 x + 2 x 2 are a + c b c . (c) For the kernel, the rowreduced matrix gives us the relations x 1 2 x 2 + 3 x 3 = 0 and x 2 x 3 = 0. x 3 is a free variable, so set x 3 = t . Then x 2 = t and x 1 = 2 t 3 t = t . Thus, ker( T ) consists of all vectors  t t t , where t is any real number. So ker( T ) = span  1 1 1 . Alternatively, this problem can be solved by inspection: if we notice the relationship among the columns: ~v 1 ~v 2 = ~v 3 , where ~v i is the i th column of the matrix for T , we obtain the same answers. (Since the rank of the matrix is 2, the ranknullity theorem tells us the nullity must be 1, so we have found eveything in the kernel) 2. (a) W is not a subspace of V . It is not closed under scalar multiplication: I = 1 0 0 1 is in W since I 2 = I . But 2 I is not in W since (2 I ) 2 = 4 I 6 = 2 I . (b) W is a subspace of V : 0 (ie. the function f ( t ) = 0) is in W : f ( t ) = 0 for all t . In particular, f (3) = 0. W is closed under scalar multiplication: If f ( t ) is in W and k is a scalar, then d ( kf ) dt = k df dt for all t . So d ( kf ) dt t =3 = k df dt t =3 = k 0 = 0. Hence, kf ( t ) is in W ....
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 Spring '06
 PANTANO
 Multivariable Calculus

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