Newberger Math 247 Spring 02
Homework solutions: Section 1.5 #26-32 and Section 1.7 #23-29 Section 1.5 #26-32. 26. Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. Ax = 0 has only the t
Newberger Math 247 Spring 03
Homework solutions: Section 1.2, #23-26, 29, 30 23. Suppose that a 3 5 coecient matrix for a system has three pivot columns. Is the system consistent? This system is consistent. The coecient matrix has 3 rows. Since the coecie
Newberger Math 247 Spring 03
Homework solutions: Section 1.4, #31-36 31. Let A be a 3 2 matrix. Explain why the equation Ax = b cannot be consistent for all b in R3 . Generalize your argument to the case of an arbitrary A with more rows than columns. Noti
Newberger Math 247 Spring 03
Homework solutions: Section 2.8 #31-36 31. Suppose F is a 5 5 matrix whose column space is not equal to R5 . What can you say about Nul F ? Start your explanations with the assumptions (the suppose part). Since the column spac
Newberger Math 247 Spring 03
Homework solutions: Section 4.1 #5-8 In Exercises 5-8 determine if the given set is a subspace of Pn for an appropriate value of n. Justify your answer. 5. All polynomials of the form p(t) = at2 . The set of all polynomials of
Newberger Math 247 Spring 02
Quiz Solutions 1.1 and 1.2 1. Consider the following system. x1 + x2 + 4 x3 = 2 2x1 2x2 + hx3 = k 2x2 + 8x3 = 0 Find all h and k such that the system has (a) no solution, (b) a unique solution, and (c) an innite number of solu
Newberger Math 247 Spring 02
Solutions for Quiz 1.3 and 1.4 Let 2 1 0 5 b = 1 u1 = 2 u2 = 1 u3 = 6 . 6 0 2 8
1. (12 points) Show that b is in Spancfw_u1 , u2 , u3 . In other words show that b is a linear combination of u1 , u2 and u3 . We want to show we
Newberger Math 247 Spring 03
Solutions for Quiz 1.5 and 1.7 1 2 1 0 2 8 10 b = 12 A= 5 4 14 9 5 2 3 8 1. (a) (8 points) Solve the system Ax = b where A and b are given above. Put your answer in parametric vector form. To solve, reduce the augmented matrix
Newberger Math 247 Spring 03
Quiz 1.8
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1. Let T be the transformation given by 1 4 A= 0 0
T (x) = Ax, where 2 8 1 0
(a) (4 points) What must a and b be so that T : Ra Rb ? The number of entries in x must match the number of columns of A for the denit
Newberger Math 247 Spring 03
Quiz 5 over Section 1.9
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1. Let T be the linear transformation that rotates clockwise by /2 radians and then reects across the x1 axis. Find the standard matrix for T . 1 0 )= , and 0 1 0 1 01 T( )= . Then the standard ma
Newberger Math 247 Spring 03
Quiz 4.1-4.2
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1. State the denition of a subspace of Rn . A subspace of Rn is any set H in Rn that has three properties: (a) The zero vector is in H . (b) For each u and v in H , the sum u + v is in H . (c) For each u in
Newberger Math 247 Spring 03
Homework solutions: Section 2.1 #18,20-24, Section 2.2 #14, 16, 18, 20 and Section 2.3 #16, 18, 22 Section 2.1 #18,20-24. 18. Suppose the rst two columns, b1 and b2 , of B are equal. What can you say about the columns of AB (i
Newberger Math 247 Spring 03
Homework solutions: Section 1.8 #26, 27, 31, Section 2.3 #34,36,37,38 Section 1.8 #26,27,31. 26. Let u and v be linearly independent vectors in R3 and let P be the plane through u, v and 0. The parametric equation of P is x =
Newberger Math 247 Spring 03
Review sheet for the Final Exam Part I This part of the nal exam will be over material covered since the second midterm. It will cover Sections 4.1-4.4, 4.7, 5.1, 5.2. You will be asked to state the denitions of the following
Newberger Math 247 Spring 03
Quiz 4.4 and 4.7
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1. Let A = cfw_a1 , a2 , a3 and B = cfw_b1 , b2 , b3 be bases for a vector space V and suppose a1 = 2b1 b2 + b3 , a2 = 3b2 + b3 , and a3 = 3b1 + b2 . a. (4 points) Find the change of coordinates matrix
Newberger Math 247 Spring 02
Sample Exam 2 1. a. (5 points) State the denition of one-to-one. A transformation T : Rn Rm is called one-to-one if for each b in Rm there is at most one x in Rn with T (x) = b. b. (5 points) State the denition of onto. A tran
Newberger Math 461 Spring 02
Sample Final Exam
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Remark: This exam does not have any questions asking the students to show they know how to show a transformation is not linear, or a subset is not a subspace. You are likely to have one or both of these
Sample Writing Assignment: Pythagorean Theorem Write about the Pythagorean Theorem.
by Gwen Fisher
First, we start with a statement of the theorem: Let ABC be a right triangle with sides of lengths a, b, and c. Let the sides with lengths a and b be the le
Warm-up We have a bag containing 9 red beans and 1 green bean and nothing else. True or False: 1. The bag contains 7 red jelly beans. True. 2. Every bean in the bag is red. False. There exist beans in the bag that are not red. 3. None of the beans in the
Match the properties to the objects that can have them.
Properties: 1. Be linearly independent. 2. Be consistent or inconsistent. 3. Have a unique solution or an infinite number of solutions. 4. Be invertible. 5. Be linear. 6. Be homogeneous. 7. Span all
Newberger Math 247 Spring 03
Review sheet for Exam 1 This exam will cover Sections 1.1 through 1.6 in Chapter 1. Use this list of questions to guide your studies. Look in the problems at the end of each sections for these questions and make sure you know
Newberger Math 247 Spring 03
Review sheet for Exam 2 This exam will cover Sections 1.8, 1.9, 2.1, 2.2, 2.3, 2.8, 3.1. You will be asked to state the denitions of the following terms. Use precise language; to be safe, you may want to use the denitions stra
Newberger Math 247 Spring 03
Exam 2 Solutions 1. a. (7 points) Calculate the determinant of the following matrix. 600 5 2 0 0 0 A= 1 7 2 5 831 8
I expanded on the second row of A. 00 5 det A = 2(1)1+2 det 7 2 5 31 8 = (2) 5(1)1+3 det = (10)(7 1 2 3) = 10.
Newberger Math 247 Spring 03
Quiz 4.1-4.2
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1. (12 points) Show that the following set is not a subspace of R3 . a+b3 a b a, b R H= 2b + 1 0 We will show that the zero vector 0 = 0 is not in H . To see if 0 is 0 in H , we will try to nd a and b such t