[4]
(b) Give an example of a subset of R2 that is closed under addition but not closed under scalar multiplication. Justify your answer.
Mห้องสมุดไป่ตู้TH 136 - Midterm
UNIVERSITY OF WATERLOO MIDTERM SPRING TERM 2012
Student Name (Print Legibly) (family name) Signature Student ID Number (given name)
COURSE NUMBER COURSE TITLE DATE OF EXAM TIME PERIOD DURATION OF EXAM NUMBER OF EXAM PAGES (Including this sheet) INSTRUCTORS/SECTIONS 001 Sean Speziale (11:30) 002 Ting Kei Pong (8:30) 003 Andrew Beltaos (12:30) EXAM TYPE ADDITIONAL MATERIALS ALLOWED
MATH 136 - Midterm
Spring Term 2012
Page 10 of 10
You may use the space below for rough work (in which case you may tear off this page), or to continue any other question that you have run out of space answering. In this case, be sure to indicate clearly, in the original location, that the work continues here.

MATH 136 - Midterm
Spring Term 2012
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[8]
7. Consider the linear system x1 + hx2 = 2 5x1 + x2 = k Find all values for h and k (if any) such that the system has (i) no solutions; (ii) a unique solution; (iii) infinitely many solutions.
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(b) With V , W , and V + W defined as in part (a), suppose that the vectors v1 , . . . , vp and w1 , . . . , wq are such that V = Span{v1 , . . . , vp } and W = Span{w1 , . . . , wq }. Prove that V + W = Span{v1 , . . . , vp , w1 , . . . , wq }.
Notes: 1. Fill in your name, ID number, section, and sign the paper. Don’t write formulas on this page. 2. Answer all questions in the space provided. The last page is for rough work. 3. Check that there are 10 sheets. 4. Your grade will be influenced by how clearly you express your ideas, and how well you organize your solutions.
MATH 136 Linear Algebra 1 For Honours Mathematics Monday, June 4, 2012 19:00 - 20:50 110 minutes 10
(please indicate your section) Closed Book NONE (NO CALCULATORS)

MATH 136 - Midterm
Spring Term 2012
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2. Consider the linear system x1 − 2x2 + x3 = 0 2x2 − 8x3 = 8 −4x1 + 5x2 + 9x3 = −9 [1] [4] [2] [1] [2] (a) Write the augmented matrix for the system. (b) Row reduce the augmented matrix into Reduced Row Echelon Form. Make sure to indicate the elementary row operations you are using. (c) Find the general solution of the system, if it exists. (d) What is the rank of the augmented matrix? (e) Find the general solution of the corresponding homogeneous system (that is, the homogeneous system with the same coefficients as the given system).
Marking Scheme: Question 1 2 3 4 5 6 7 8 Total Mark Out of 10 10 11 9 8 9 8 10 75
MATH 136 - Midterm
Spring Term 2012
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3 3 1. Let v = 0 and w = 3 . 0 0 [1] [3] [3] [3] (a) Find ||w||. (b) Find the angle between the vectors v and w (you may give your answer in radians or degrees). (c) Find v × w. (d) Find projw (v ).
Spring Term 2012
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[3]
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1 −1 6. Let a = 1 and b = 2 . −2 3 −1 (a) Let x = 5 . Is x ∈ Span{a, b}? If so, find the coefficients that make x a linear 5 combination of a and b. −1 (b) Let y = 5 . Is y ∈ Span{a, b}? If so, find the coefficients that make x a linear 4 combination of a and b. −2 (c) Let c = 10 . Is the set{a, b, c} linearly independent? Prove why or why not. 8
MATH 136 - Midterm
Spring Term 2012
Page 6 of 10
[4]
5. (a) Let S = {v1 , v2 , v3 } and assume that S is an orthogonal set. Prove that ||v1 + v2 + v3 ||2 = ||v1 ||2 + ||v2 ||2 + ||v3 ||2 . Hint: Recall that ||x||2 = x · x.
MATH 136 - Midterm
Spring Term 2012
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3. Let P be the plane in R3 containing the points A = (0, 1, 1), B = (1, −1, 1), and C = (1, 0, 1). Let L be the line perpendicular to P , containing the point A. Let U = (1, −1, 0). Find the following: [4] [3] [4] (a) a scalar equation for the plane P . (b) a vector equation for the line L. (c) The point on the line L which is closest to the point U .
MATH 136 - Midterm
Spring Term 2012
Page 9 of 10
[5] [5]
8. (a) Prove that any set which is the span of some vectors in Rn is a subspace of Rn . That is, let S = Span{v1 , . . . , vk }, where v1 , . . . , vk ∈ Rn . Prove that S is a subspace of Rn . (b) Prove the following statement: The set {v1 , v2 } is linearly dependent if and only if one of the vectors is a scalar multiple of the other.