Math 51 First Exam October 19, 2017

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1 Math 5 First Exam October 9, 27 Name: SUNet ID: ID #: Complete the following problems. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify your answers unless specifically instructed to do so. You may use any result proved in class or the text, but be sure to clearly state the result before using it, and to verify that all hypotheses are satisfied. Please check that your copy of this exam contains 2 numbered pages and is correctly stapled. This is a closed-book, closed-notes exam. No electronic devices, including cellphones, headphones, or calculation aids, will be permitted for any reason. You have 2 hours. Your organizer will signal the times between which you are permitted to be writing, including anything on this cover sheet, and to have the exam booklet open. During these times, the exam and all papers must remain in the testing room. When you are finished, you must hand your exam paper to a member of teaching staff. Paper not provided by teaching staff is prohibited. If you need extra room for your answers, use the extra space provided at the front of this packet, and clearly indicate that your answer continues there. Do not unstaple or detach pages from this exam. It is your responsibility to look over your graded exam in a timely manner. You have only until Thursday, November 2, to resubmit your exam for any regrade considerations; consult your section leader about the exact details of the submission process. Please sign the following: On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination. Signature:

2 Math 5, Autumn 27 First Exam October 9, 27 Page of 2. (9 points) For this problem, let P be the plane in R 3 containing the points (, 9, 2), (, 7, 2), (,, 2). (a) Find, showing your steps, a nonzero vector that is normal to P. (b) Suppose A is the point (,, ); it is a fact, which you do not have to prove, that A does not lie on the plane P. Give, with reasoning, an example of a line that passes through A but does not intersect the plane P; express your line in parametric form.

3 Math 5, Autumn 27 First Exam October 9, 27 Page 2 of 2 (c) Suppose L is the line you gave in (b) on the facing page. Find, with reasoning, the length of the shortest line segment joining a point on L and a point on the plane P. (Hint: You may use, without proof, that such a segment exists and must be perpendicular to L and P; note that it can t have length zero since L and P do not intersect.)

4 Math 5, Autumn 27 First Exam October 9, 27 Page 3 of 2 2. (2 points) (a) Complete the following sentence: a set {w,..., w k } of vectors in R n is defined to be linearly dependent if (b) Suppose {u, v, w} is a linearly independent set of vectors in R n. Determine whether the set {u v, 2v w, w 2u} is linearly dependent or independent. Give complete reasoning.

5 Math 5, Autumn 27 First Exam October 9, 27 Page 4 of 2 For parts (c) (e), give explicit examples (in R 3 ) of each of the following, or, if an example does not exist, please write impossible. No justification is required for any of these parts. (c) A linearly dependent set of three vectors in R 3, but with no two vectors parallel to each other. (d) A linearly dependent set {v, v 2, v 3 } of three vectors in R 3, but for which v does not lie in span(v 2, v 3 ). (e) A linearly dependent set {v, v 2, v 3 } of three vectors in R 3, but for which v does not lie in span(v 2, v 3 ) and v 2 does not lie in span(v, v 3 ).

6 Math 5, Autumn 27 First Exam October 9, 27 Page 5 of 2 3. ( points) Suppose all we know about the 4 6 matrix A is the following information: A = a a 2 a 3 a 4 a 5 a and rref(a) =. Using this information, for parts (a) and (b) specify each of the following as completely as you can (expressing in terms of the vectors a,..., a 6 R 4 if necessary), showing all your reasoning: (a) A basis for N(A), the null space of A: (b) A basis for C(A), the column space of A: (c) Fill-in each (no justification needed): Rank of A: Nullity of A: (d) For this part, suppose we know additionally that a = [ ] and a 5 = [ ]. In the list below, place a box around each of the columns of A that we must know completely (you do not need to justify, nor provide any computed entries): a a 2 a 3 a 4 a 5 a 6

7 Math 5, Autumn 27 First Exam October 9, 27 Page 6 of 2 4. (9 points) (a) Each of the statements below about two vectors x, y R 5 is either always true ( T ), or always false ( F ), or sometimes true and sometimes false, depending on the situation ( MAYBE ). For each statement, decide which and circle the appropriate choice; you do not need to justify your answers. Given x = 2 and x y = 3, then y = 2. T F MAYBE Given x = 2 and x y = 3, then y = 3 2. T F MAYBE Given x = 2 and x y = 3, then y =. T F MAYBE Given x = 2 and x y =, then span(x, y) is a 2-dimensional subspace of R 5. T F MAYBE Given x = 2 and x y =, then span(x, y) is a 2-dimensional subspace of R 5. T F MAYBE (b) Now suppose that v, w are vectors of length (i.e., unit vectors) in R 5, and also that v w =. Given all of this information, which of the following vectors must also have length? Circle one or more. You do not need to justify your answer(s). v v w v + w v + 2w

8 Math 5, Autumn 27 First Exam October 9, 27 Page 7 of (9 points) Suppose v =, v 2 =, v 3 3 = 3, v 4 4 = (a) Find, showing your steps, conditions on the entries of b = b b 2 b 3 b 4 b 5 that determine exactly when b can be expressed as a linear combination of v, v 2, v 3, v 4. Your answer should consist of one or more linear equations involving the entries of b. (b) Express v 4 as a linear combination of v and v 3, or explain why this is not possible; show all reasoning.

9 Math 5, Autumn 27 First Exam October 9, 27 Page 8 of 2 6. (9 points) Suppose that all we know about a certain matrix A is that the set of solutions to the system of equations Ax = is a set, called S, that can be drawn in the plot below in R 2 : S 2 (a) Let T be the set of solutions to the system Ax = 2. Which of the following statements 2 is true? Circle one no justification needed, but if you circle choice (i), you must also sketch T above: i. T is drawn and labeled in my diagram above. ii. T can t be drawn because it is an empty set. iii. T can t be drawn because it is all of R 2. iv. There is not enough information provided to determine T. (b) Let W be the set of solutions to the system Ax =. Which of the following statements is true? Circle one no justification needed, but if you circle choice (i), you must also sketch W above: i. W is drawn and labeled in my diagram above. ii. W can t be drawn because it is an empty set. iii. W can t be drawn because it is all of R 2. iv. There is not enough information provided to determine W. (c) Let P be the set of solutions to the system Ax = 2. Which of the following statements is true? Circle one no justification needed, but if you circle choice (i), you must also sketch P above: i. P is drawn and labeled in my diagram above. ii. P can t be drawn because it is an empty set. iii. P can t be drawn because it is all of R 2. iv. There is not enough information provided to determine P.

10 Math 5, Autumn 27 First Exam October 9, 27 Page 9 of 2 7. ( points) (a) Complete the following: A set V of vectors in R n is defined to be a (linear) subspace of R n if (b) Suppose W is the set of vectors x = [ x 2 3 x 4 ] in R 4 having the property that x = 3x 3 and x 2 = x + 5x 4. Explain completely why W is a subspace of R 4.

11 Math 5, Autumn 27 First Exam October 9, 27 Page of 2 (Problem continued; see information on facing page) 3 (c) Find, with complete reasoning, a basis for W that contains the vector 2, or explain why such a basis does not exist.

12 Math 5, Autumn 27 First Exam October 9, 27 Page of 2 8. (2 points) Suppose all you have been told about a certain matrix A is that: A has size 3 4; and the equation Ax = does not have any solutions x; and the equation Ax = has at least one solution x. For each of parts (a) (d), determine the number of solutions x to the given equation involving this matrix A. If it is impossible to completely determine this from the given information, say so. Briefly justify your answers. (Note: examples of possible responses would be No solution, or Exactly one solution, or Infinitely many solutions, or At least one solution, or At most three solutions, and so forth, plus appropriate justification. Note that some of these phrases are more precise than others; be as precise as possible in each case.) (a) Ax = (b) Ax =

13 Math 5, Autumn 27 First Exam October 9, 27 Page 2 of 2 (Problem continued; see information on facing page) (c) Ax = (d) Ax = 2

Math 51 Midterm 1 July 6, 2016

Math 51 Midterm 1 July 6, 2016 Name: SUID#: Circle your section: Section 01 Section 02 (1:30-2:50PM) (3:00-4:20PM) Complete the following problems. In order to receive full credit, please show all of your