MAT22A_MT2_Practice_Puckett_2_KEY

MAT22A_MT2_Practice_Puckett_2_KEY - Math 22A Solutions to...

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Math 22A Solutions to Practice Midterm 02 March 01, 2012 Name: Student ID Number: Read each problem carefully. Write each step of your reasoning clearly. The best strategy is to solve the easiest problem first, the second easiest problem next, etc., working your way up to the problems you find to be most difficult. Note that in order to employ this strategy, you must read all of the problems first before working on any one of them. This is a closed-book exam. You may not use the textbook, crib sheets, notes, or any other outside material. Do not bring your own scratch paper. Do not bring blue books. No calculators, laptop computers or cell phones are allowed during the exam. The exam is to test your basic understanding of the material. Everyone is expected to work on their own exam. Any suspected acts of collaboration, copying, or other violations of the Student Code of Conduct will be brought to the attention of the Student Judicial Board. Problem Score 1. (20 points) 2. (10 points) 3. (20 points) 4. (20 points) 5. (30 points) 6. (20 points) Extra Credit Total Thursday 1 st March, 2012 at 12:09 – 0 –
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Math 22A Solutions to Practice Midterm 02 March 01, 2012 Problem 01 (20 points) Let V be a vector space and let W V be a subset of V . (a) (5 points) Define what it means for W to be a subspace of V . There are two equivalent definitions for W V to be a subspace of V . Stating either of these definitions constitutes a valid answer. Definition I: W V is a subspace of the vector space V if and only if for all w 1 , w 2 W and all scalars c R (i) w = w 1 + w 2 W and (ii) w = c w 2 W . Definition II: W V is a subspace of the vector space V if and only if for all w 1 , w 2 W and all scalars c 1 , c 2 R w = c 1 w 1 + c 2 w 2 W . (b) (5 points) Let A be an arbitrary m × n matrix. Define N ( A ), the nullspace of A . N ( A ) ≡ { x | A x = 0 } (1) Thursday 1 st March, 2012 at 12:09 – 1 – Score for this page:
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Math 22A Solutions to Practice Midterm 02 March 01, 2012 (c) (5 points) What vector space V contains N ( A ). In other words, find V such that N ( A ) V . Give an explanation for your answer. Since A is an m × n matrix the domain of A is R n . Therefore, given the definition of the nullspace of A in equation (1) in Problem 01(b) above, N ( A ) R n . (d) (5 points) Is N ( A ), the nullspace of A , a subspace of V ? If the answer is “yes”, show that all of the conditions you listed in Problem 2(a) above hold. If the answer is “no”, give an example of how the conditions in Problem 2(a) above fail to hold. The the nullspace of A is a subspace of R n , the domain of A . Proof: We will show that N ( A ) satisfies the condition in Definition II in part (a) above. Let x 1 , x 2 N ( A ) be any two vectors in N ( A ) and let c 1 , c 2 R be any two scalars in R . We must show that x = c 1 x 1 + c 2 x 2 N ( A ) .
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