MATH_215_Test1 - Practice for Test 1 For full credit, show...

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Unformatted text preview: Practice for Test 1 For full credit, show all work. 1 State the precise definition of the following concepts: (a) span of a set of vectors in Rn . (b) Linear combination. 1 1 0 2 Let u = 2 , v = 1 , w = 1 3 0 1 (a) Compute the distance and angle between u and v. (b) Find projv (u) (c) Find the distance between the parallel planes 3x - y + 2z = 7 and 3x - y + 2z = 2. Note that u = (1, 2, 3) lies on the first plane and v = (1, 1, 0) lies on the second plane. (d) Show that u, v, w are linearly independent. 3 Consider the following system of linear equations: x - 2y + z =2 ty + z =3 (t - 1)z =t2 - 1 where t is a parameter. (a) For which t is the above system inconsistent, has unique solution or has infinite number of solutions. 2 (b) Use the above system of linear equations to express 3 as a linear 3 1 -2 1 combination of 0 , 2 , 1. 0 0 1 4 Consider a row of three lights, each of which can be off, dim and bright, cycling in this order of description. Associated with each light is a button. Pressing a button changes the states of the light associated with the button and the adjacent light(s). Suppose the first and third lights are initially dim and the second light is off. Is it possible to have the first light bright and the second and third lights off? If so, describe what needs to be done to obtain such a configuration. 5 Prove 2 of the following statements (complete the remaining one problem for extra credit): (a) Show that the distance between parallel planes with equations n x = |d1 - d2 | d1 and n x = d2 is given by . ||n|| (b) Let {v1 , , vk } be linearly independent set of vectors in Rn and let v Rn . Suppose v = c1 v1 + + ck vk with c1 = 0. Show that {v, v2 , , vk } is linearly independent. (c) Prove that proju (proju (v)) = proju (v). ...
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MATH_215_Test1 - Practice for Test 1 For full credit, show...

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