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Course: CIS 120, Fall 2009
School: UPenn
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Word Count: 1508

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dispatch Dynamic again Static members Extra office hours this week Extra office hours this week Brent: Wednesday, Moore 100A it is, 5-7PM Alexey: Thursday 5-7, location TBA No office hours for Brent on Friday 9/23/09 2 Sample Exams Posted on class web site 9/23/09 3 HW2 9/23/09 4 Inheritance of fields public class Item { protected Container container; protected String description; ... } public...

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Waterloo - MATH - 136
Math 136Assignment 10 Solutions1. By checking whether columns of P are eigenvectors of A, determine whether P diagonalizes A. If so, determine P 1 , and check that P 1 AP is diagonal. a) A = 42 13 ,P= . 5 3 1 1Solution: Check whether the columns of P a
Waterloo - MATH - 136
Math 136Assignment 9 Solutions1. Calculate the determinant of the following matrices. 0 1 1 0 3 0 0 2 a) A = 0 1 2 1 50 0 7 Solution: Expand along the fourth row: 0 3 det A = 0 5 1 1 00 12 00 0 1 1 0 1 1 0 1 1 1 1 2 (5)(2) = 3 1 2 1 5 0 0 2 = (3)(7) 12
Waterloo - MATH - 136
Math 136Assignment 8 Solutions1. For each of the following matrices, nd the inverse, or show that the matrix is not invertible. 1 1 2 a) A = 3 1 5. 223 Solution: To determine if A is 1 1 3 1 22 Since the RREF 1 1 0 1 b) B = 2 2 10 invertible we write 21
Waterloo - MATH - 136
Math 136Assignment 7 Solutions1. Show the each of the following sets form a basis for the subspace that they span, and determine the coordinates of x and y with respect to the basis. a) cfw_(1, 1, 0, 1, 0), (1, 0, 2, 1, 1), (0, 0, 1, 1, 3); x = (2, 2, 5
Waterloo - MATH - 136
Math 136Assignment 6 Solutions1. Determine, with proof, which of the following are subspaces of the given vector space. Find a basis for each subspace. Solution: A is the solution space of a homogeneous system of linear equations 2x1 + x3 = 0, x1 + x2 x
Waterloo - MATH - 136
Math 136Assignment 5Due: Wednesday, Feb 24th1. Let A = a) 2A B1 2 1 10 0 5 1 2 ,B= ,C= . Determine the following 3 2 1 0 2 3 1 1 2 2 4 2 10 0 1 4 2 = 6 4 2 0 2 3 6 6 1 1 6 1 1 = 1 1 . 2 2 1 1 24 1 = 18 2 1 2 18 . 42Solution: 2A B = b) A(B T + C T )
Waterloo - MATH - 136
Math 136Assignment 4 Solutions 3 2 1 2 1 0 1 0 1 0 , C = 1 . 1. Let A = ,B= 2 1 2 13 3 2 Solution: a) AB = 2 1 4 . 4 1 8Determine the following products or state that they are undened.b) BA is undened since B has 3 columns but A has only 2 rows. c) AC
Waterloo - MATH - 136
Math 136Assignment 3 Solutions1. For each of the following systems of linear equations: i) Write the augmented matrix. ii) Row-reduce the augmented matrix into row echelon form. iii) Find the general solution of the system or explain why the system is i
Waterloo - MATH - 136
Math 136 1. DetermineAssignment 2 Solutionsa) proj(2,1,1) (3, 2, 1) and perp(2,1,1) (3, 2, 1). Solution: (3, 2, 1) (2, 1, 1) 9 (2, 1, 1) = (2, 1, 1) = (3, 3/2, 3/2) (2, 1, 1) 2 6 perp(2,1,1) (3, 2, 1) = (3, 2, 1) proj(2,1,1) (3, 2, 1) = (3, 2, 1) (3, 3/
Waterloo - MATH - 136
Math 136Assignment 1 Solutions1. Compute each of the following. a) (1, 3, 4) + (1, 1, 2) Solution: (1, 3, 4) + (1, 1, 2) = (0, 4, 6). b) 3(1, 1, 2) 2(2, 0, 3).Solution: 3(1, 1, 2) 2(2, 0, 3) = (3, 3, 6) + (4, 0, 6) = (7, 3, 12). Solution: The distance
Waterloo - MATH - 136
Math 136Assignment 10Not To Be Handed In1. By checking whether columns of P are eigenvectors of A, determine whether P diagonalizes A. If so, determine P 1 , and check that P 1 AP is diagonal. a) A = b) A = 42 13 ,P= . 5 3 1 1 13 11 ,P= . 31 1 12. Let
Waterloo - MATH - 136
Waterloo - MATH - 136
Math 136Assignment 8Due: Wednesday, Mar 24th1. For each of the following matrices, nd the inverse, or show that the matrix is not invertible. 1 1 0 2 1 1 2 0 1 1 0 3 1 5. b) B = a) A = 2 2 3 5. 223 1 0 13 2 1 1 1 . Find B 1 and use it to solve B x = d,
Waterloo - MATH - 136
Math 136Assignment 7Due: Wednesday, Mar 10th1. Show the each of the following sets form a basis for the subspace that they span, and determine the coordinates of x and y with respect to the basis. a) cfw_(1, 1, 0, 1, 0), (1, 0, 2, 1, 1), (0, 0, 1, 1, 3
Waterloo - MATH - 136
Waterloo - MATH - 136
Math 136Assignment 5Due: Wednesday, Feb 24th1. Let A = a) 2A B1 2 1 10 0 5 1 2 ,B= ,C= . Determine the following 3 2 1 0 2 3 1 1 2 b) A(B T + C T ) c) BAT + CAT2. Prove that if x M (3, 2) and a, b R are scalars, then (a + b)x = ax + bx. 3. Determine
Waterloo - MATH - 136
Math 136Assignment 4Due: Wednesday, Feb 3rd 3 2 1 2 1 0 1 0 1 0 , C = 1 . 1. Let A = ,B= 2 1 2 13 3 2 a) AB b) BA c) AC d) B T CDetermine the following products or state that they are undened. e) C T C f) BAT2. If AB is a 2 4 matrix, then what size a
Waterloo - MATH - 136
Waterloo - MATH - 136
Waterloo - MATH - 136
Math 136Assignment 1Due: Wednesday, Jan 13th1. Compute each of the following. a) (1, 3, 4) + (1, 1, 2) b) 3(1, 1, 2) 2(2, 0, 3).2. Determine the distance between P (2, 1, 1) and Q(1, 1, 1). 3. Determine which of the following pairs of vectors is ortho
Waterloo - MATH - 136
Math 136Sample Term Test 1NOTES: - Questions 4d, 5, 6 on this test cover material that will not be covered on our term test 1. - Students had 90 minutes to write this test, where you will have 110 minutes. 1. Short Answer Problems a) List the 3 elementa
Waterloo - MATH - 136
Math 136Term Test 1 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) List the 3 elementary row operations. Solution: 1. Multiply
Waterloo - MATH - 136
Math 136Sample Term Test 1 - 2NOTES: - Questions 7, 10b on this test cover material that will not be covered on our term test 1. 1. Short Answer Problems a) List the 3 elementary row operations. b) What can you say about the consistency and the number o
Waterloo - MATH - 136
Math 136Sample Term Test 1 - 2NOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) List the 3 elementary row operations. Solution: 1. Multip
Waterloo - MATH - 136
Math 136Sample Term Test 2 # 1NOTES: - In addition to these questions you should also do questions 4d, 5, 6 from sample term test 1 # 1. 1. Short Answer Problems a) What is the denition of the row space and column space of a matrix A. b) What is the den
Waterloo - MATH - 136
Math 136Sample Term Test 2 # 1 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) What is the denition of the row space and column
Waterloo - MATH - 136
Math 136Sample Term Test 2 # 2NOTES: - In addition to these questions you should also do questions 7, 10 b from sample term test 1 # 2. 1. Short Answer Problems a) Let S = cfw_v1 , . . . , vn be a non-empty subset of a vector space V . Dene the stateme
Waterloo - MATH - 136
Math 136Sample Term Test 2 # 2 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Let S = cfw_v1 , . . . , vn be a non-empty subse
Waterloo - MATH - 136
Math 136 1. Short Answer ProblemsTerm Test 1 Solutions[2] a) Calculate proj(1,1,2) (1, 2, 2). Solution: proj(1,1,2) (1, 2, 2) =(1,2,2)(1,1,2) (1, 1, 2) (1,1,2) 2= 5 (1, 1, 2). 6[1] b) If n = a b, then what is a n?Solution: Since a b gives an vector
Waterloo - MATH - 136
Math 136Term Test 2 InformationMonday, Mar 15th, 7:00 - 8:50 pmMaterial Covered: Sections 3-1, 3-2, 3-3, 3-4, 4-1, 4-2, 4-3, 4-4. You need to know: - All denitions and statements of theorems and lemmas. - How to prove a subset of a vector space is or i
Waterloo - MATH - 136
Math 136 1. Short Answer ProblemsTerm Test 2 Solutions[1] a) What is the denition of a basis B of a vector space V ? Solution: A basis is a linearly independent spanning set. [1] b) What is the denition of the dimension of a vector space V ? Solution: T
Waterloo - MATH - 136
Math 136Final Exam InformationWednesday, April 14, 2010 9:00 AM - 11:30 AM PAC 1,2,3,4,5,6,7,8,11,12Material Covered: Entire Course. Information: - Although the test covers the entire course, there is an emphasis on material after term test 2. - You ne
Waterloo - MATH - 136
Math 136 - Final Exam Winter 2009NOTE: The questions on this exam does not exactly reect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not a
Waterloo - MATH - 136
NOTE: These are only answers to the problems and not full solutions! On the nal exam you will be expected to show all steps used to obtain your answer. 1. Short Answer Problems a) det A = 3 (2) = 5 so A1 = 1 0 0 b) L1 = 0 1 0 , 0 01 3 2 . 11 10 0 1 0 0 L2
Waterloo - MATH - 136
Math 136 - Final Exam Spring 2009NOTE: The questions on this exam does not exactly reect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not a
Waterloo - MATH - 136
1. a) If the rank is 4 and there are 4 linear equations, then there is a leading 1 in each row of the coecient matrix and hence the system is consistent for all b. The number of parameters in the general solution is 5-4=1. b) v = 1(x2 + 1) + 2(x2 + 2x) +
Waterloo - MATH - 136
Math 136Tutorial 1 Problems1: Let x = (3, 5, 1) and y = (2, 1, 1). Calculate 2x 3y , (x + y ) x and determine if x and y are orthogonal. 2: Verify the Cauchy-Schwarz inequality and the triangle inequality if x = (2, 6, 3) and y = (3, 4, 5). 3: Find an e
Waterloo - MATH - 136
Math 136Tutorial 2 Problems1: Determine proj(1,0,1) (6, 2, 6) and perp(1,0,1) (6, 2, 6). 2: Find the distance from the point P (0, 2, 1) to the plane 2x1 x3 = 5. 3: Calculate the area of the parallelogram determined by (2, 3) and (5, 2) in two dierent w
Waterloo - MATH - 136
Math 136Tutorial 3 Problems 1 0 1 1: Consider S = 2 , 1 , 1 . Determine which of the following are in the span 4 1 1 of S . Write each vector in the span as a linear combination of the vectors in S . 2 a) a = 7. 5 1 19 . b) b = 17 c) Is span S = R3 ? Is
Waterloo - MATH - 136
Math 136Tutorial 4 Problems 12 3 1 3 1: Let A = 3 1, B = ,C= . 1 1 4 21 a) AB b) CB c) C T AT .Calculate the following or state that they dont exist:2: Let S = cfw_(1, 1, 1), (0, 1, 1), (2, 1, 1), (1, 2, 3).a) Write the system of linear equations we
Waterloo - MATH - 136
Math 136Tutorial 5 Problems1: a) Let S : R2 R2 be a stretch by a factor of 5 in the x2 -direction. What is the standard matrix of S . b) Calculate the standard matrix of the composition of S followed by a rotation through angle . 3 c) Calculate the stan
Waterloo - MATH - 136
Math 136Tutorial 6 Problems1: Determine, with proof, which of the following are subspaces of R4 . a) The solution space of a homogeneous system with 3 equations in 4 unknowns. b) W = x1 x2 M (2, 2) | x1 + x2 + x3 + x4 = 1 . x3 x4c) S = cfw_p(x) P2 | p(
Waterloo - MATH - 136
Math 136Tutorial 7 Problems1: a) Verify that B = cfw_(1, 0, 1), (1, 1, 2), (1, 1, 1) is a basis for R3 . 1 2, what is v ? b) If [v ]B = 3 2: Find a basis and determine the dimension ofc) Determine the coordinates of x = (1, 1, 1) and y = (4, 2, 7) with
Waterloo - MATH - 136
Math 136Tutorial 8 Problems 1 2 1 3 5 7 1. 1: Find the inverses of A = and B = 3 2 2 1 2 1 2: Let A = 13 . Write A and A1 as a product of elementary matrices. 2 4 2 5 5 2 , and use it to solve Ax = . 1 0 2 133 7 3 5 3: Find an LU-factorization of A = 6
Waterloo - MATH - 136
Math 136Tutorial 9 Problems1: Find the determinant of the following matrices: 123 4 1 2 1 1 1 1 2 1 2 2 2 3 , b) B = a) A = 2 1 2 1 2 1 01 1 1 3 1 3 6 3 1 2: Use Cramers rule to solve x1 5x2 2x3 = 2 2x1 + 3x3 = 3 4x1 + x2 x3 = 1 3: Prove that det A1 =1
Waterloo - AFM - 101
Waterloo - AFM - 101
Waterloo - AFM - 101
Waterloo - AFM - 101
Waterloo - AFM - 101
Waterloo - AFM - 101
Waterloo - AFM - 101
Waterloo - AFM - 101
Waterloo - AFM - 101
Waterloo - AFM - 101
Waterloo - AFM - 101
Keller Graduate School of Management - ACCT - AC557
EXERCISE 12-1 (1520 minutes) (a) (b) 10, 13, 15, 16, 17, 19, 23 1. Long-term investments in the balance sheet. 2. Property, plant, and equipment in the balance sheet. 3. Research and development expense in the income statement. 4. Current asset (prepaid r
Keller Graduate School of Management - ACCT - ACC557
EXERCISE 13-1 (1015 minutes)(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) Current liability. Current liability. Current liability or long-term liability depending on term of warranty. Current liability. Footnote disclosure (assume not p
Keller Graduate School of Management - ACCT - ACC557
EXERCISE 13-1 (1015 minutes)(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) Current liability. Current liability. Current liability or long-term liability depending on term of warranty. Current liability. Footnote disclosure (assume not p
Keller Graduate School of Management - ACCT - ACC552
EXERCISE 13-4 (2025 minutes) SANTANA COMPANY Partial Balance Sheet December 31, 2010 Current liabilities: Notes payable (Note 1). Long-term debt: Notes payable expected to be refinanced in 2011 (Note 1). \$4,000,000*3,000,000Note 1. Under a financing agr