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e508fk - because the scores were fairly high on Q5 the...

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MAS 3105 Nov 7, 2008 Quiz 5 and Key Prof. S. Hudson 1) Suppose that L ((1 , 2) T ) = (5 , 11) T and L ((3 , 4) T ) = (11 , 25) T . Find L ((4 , 6) T ). 2) Find the transition matrix from [ v 1 , v 2 , v 3 ] to [ u 1 , u 2 , u 3 ] where: v 1 = (4 , 6 , 7) T , v 2 = (0 , 1 , 1) T , v 3 = (0 , 1 , 2) T u 1 = (1 , 1 , 1) T , u 2 = (1 , 2 , 2) T , u 3 = (2 , 3 , 4) T 3) Choose ONE of these to prove (on the back of the page). a) If A is similar to B then A T is similar to B T . b) Suppose that L : V W is linear. Prove that Ker ( L ) is a subspace of W . c) The dimension of Col(A) equals the dimension of Row(A). Bonus (about 5 pt): Give an example of a linear transformation which does not have a matrix representation. Remarks and Answers: The average grade was about 50/60. You can use this (unoffi- cial) scale: A’s = 54 to 60 , B’s = 48 to 53 , C’s = 43 to 38 ,D’s = 33 to 37. You can estimate your semester grade using the same scale. Average your best 4 of 5 quiz grades. Compare that number with the scale above. The class average for this number is also about 50/60. This is a bit inflated, because I just dropped a grade and
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Unformatted text preview: because the scores were fairly high on Q5; the scale will probably come down a little with Q6 and the final. Of course, I will also include your HW and MHW later on. 1) Notice that (4 , 6) T = (1 , 2) T + (3 , 4) T (this made the problem fairly easy - otherwise you’d have to solve a linear system (or compute a transition matrix) to find the right LC). So, L ((4 , 6) T ) = L ((1 , 2) T ) + L ((3 , 4) T ) = (16 , 36) T . 2) V U-1 = 1-1 2 1 1 1 1 3) Parts b) and c) are in the text/lectures, but part a) is easiest. B) Any example of L : V → W in which V (or W ) is infinite-dimensional. In that case, the v ∈ V can’t be represented as column vectors. One example is d/dx : P → P . 1...
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