Math 221 Midterm Fall 2017 Section 104 Dijana Kreso

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1 The University of British Columbia Midterm October 5, 017 Group B Math 1: Matrix Algebra Section 104 (Dijana Kreso) Last Name: Student Number: First Name: Section: Format: 50 min long exam. Total: 5 marks. Worth 30% of your final mark. part A: 10 short questions worth 1 mark each part B: 3 long questions worth 5 marks each Rules governing examinations: 1. No books, notes, electronic devices or any papers are allowed.. You must be prepared to produce your library/ams card upon request. 3. You are not allowed to communicate with other students during the examination. 4. You are not allowed to purposely view other s written work nor purposefully expose your own work to the view of others. 5. You must follow all instructions provided by the invigilators. 6. Any deviation from these rules will be treated as an academic misconduct. The plea of accident or forgetfulness shall not be received. I agree to follow the rules outlined above (signature) Question A1 A10 B1 B B3 TOTAL Points Score 1

2 Part A - Short Answer Questions, 1 mark each. A1: Consider a system of two equations x + y + kz = and x + y + 3z = 1. For which k is the system consistent? A: Let e 1, e, e 3 be the standard vectors in R 3, that is 1 e 1 = 0, 0 e = 1, 0 e 3 = 0, and let f 1 = e 1 + e, f = e 1 e + e 3, f 3 = e e 3. Which of the following sets is linearly independent? (Only one correct answer.) (a) {f 1, f, f 3 } (b) {e 1, f, f 3 } (c) {e, e 3, f 3 }

3 A3: Let H be the solution set of a system x + y + z = 1, x + y z = and x + y + 4z =. Is H a plane, a line, a point or an empty set? A4: Let A denote the augmented matrix for a system of 5 linear equations in 6 variables. Which of the following is true? (a) A is a 6 6 matrix. (b) A is a 5 7 matrix. (c) A is a 5 6 matrix. A5: Circle all of the transformations below that are linear. (a) T : R 3 R 3 given by T (x 1, x, x 3 ) = (0, x, 3x 3 ). (b) T : R 3 R 3 given by T (x 1, x, x 3 ) = (x 1 x, x x 3, x 3 x 1 ). (c) T : R 3 R 3 given by T (x 1, x, x 3 ) = (x 1, 1, 1). (d) T : R 3 R 3 given by T (x 1, x, x 3 ) = (x 1, x, x + x 3 ). 3

4 A6: Compute the following product A7: Circle all the sets below which are not subspaces of R 3? (a) The plane x + 3z = 0. (b) The intersection of planes x + y + 3z = 1 and x + y z = 0. (c) The intersection of planes x + y + 3z = 1 and x + y z = 1. (d) The intersection of a plane x + y z = 0 and the line through (0, 0, 0) and ( 1,, ). A8: Let A be an invertible n n matrix. Circle all true statements below. (a) The equation Ax = b has solutions for infinitely many b R n. (b) The null space of A contains a finite number of vectors. (c) If AB = BA for a matrix B, then B = A 1. (d) The set of vectors x R n such that Ax = b is a subspace of R n for infinitely many b R n. 4

5 A9: Let k be a real number. Consider the matrix below. [ ] 1 k A = k 4 Circle all true statements below. (a) The rows of A are linearly independent for all values of k. (b) The determinant of A is always nonzero. (c) The columns of A span R if and only if k is not equal to. (d) The equation Ax = 0 has solutions for infinitely many values of k. A10: Circle all of the matrices below that are invertible? (a) (b) (c) (d) 3 M = M = M = M =

6 Part B - Long Answer Questions, 5 marks each. B1: Consider the system of linear equations below in variables x 1, x, x 3. x 1 + x = 1 x 1 + x 3 = x + x 3 = 3 (a) [1 mark] Write the augmented matrix of the system. (b) [ marks] Find reduced echelon form of the augmented matrix using row-reduction procedure. (c) [1 mark] Find the general solution of the system in parametric vector form. (d) [1 mark] Find the solution of the system with x 1 = 100. Show your work. 6

7 B: Consider the linear transformations below. For each one write down its standard matrix. Show your work. a) [1 mark] T : R R reflects every vector through the line y = x and multiplies the resulting vector by 3. b) [ marks] T : R 3 R 3 defined by T 1 = 1, T 1 = 1, T = c) [ marks] Are the linear transformations above onto? Are they one-to-one? Explain. 7

8 B3: Let k be a real number and 1 k 1 A = k (a) [1 mark] Show that A is invertible for any value of k. (b) [ marks] Find B = A 1. (Your answer will depend on k). (c) [ marks] Calculate B 5 A (B 1 ) 4. Show your work. 8

9 The University of British Columbia Midterm October 5, 017 Group A Math 1: Matrix Algebra Section 104 (Dijana Kreso) Last Name: Student Number: First Name: Section: Format: 50 min long exam. Total: 5 marks. Worth 30% of your final mark. Part A: 10 short questions worth 1 mark each. Part B: 3 long questions worth 5 marks each. Rules governing examinations: 1. No books, notes, electronic devices or any papers are allowed.. You must be prepared to produce your library/ams card upon request. 3. You are not allowed to communicate with other students during the examination. 4. You are not allowed to purposely view other s written work nor purposefully expose your own work to the view of others. 5. You must follow all instructions provided by the invigilators. 6. Any deviation from these rules will be treated as an academic misconduct. The plea of accident or forgetfulness shall not be received. I agree to follow the rules outlined above (signature) Question A1 A10 B1 B B3 TOTAL Points Score 1

10 Part A - Short Answer Questions, 1 mark each. A1: Let A denote the augmented matrix for a system of 4 linear equations in 3 variables. Which of the following is true? (a) A is a 3 4 matrix. (b) A is a 4 3 matrix. (c) A is a 4 4 matrix. A: Consider a system of two equations x + y = 0 and x + y + kz = 1. For which k is the system consistent? A3: Let H be the solution set of a system of linear equations x + y + z = 1 and x + y z = 0. Is H a plane, a line, a point or an empty set?

11 A4: Let e 1, e, e 3 be the standard vectors in R 3, that is 1 e 1 = 0, 0 e = 1, 0 e 3 = 0, and let f 1 = e 1 + e, f = e 1 e + e 3, f 3 = e 1 + e 3. Which of the following sets is linearly independent? (Only one correct answer.) (a) {f 1, f, f 3 } (b) {e 1, e 3, f 3 } (c) {e 1, f, f 3 } A5: Let k be a real number. Consider the matrix below. [ ] k 9 A = 1 k Circle all true statements below. (a) A is invertible for infinitely many values of k. (b) The determinant of A is always nonzero. (c) The columns of A span R if and only if k is not equal to 3. (d) The equation Ax = 0 has a unique solution for infinitely many values of k. 3

12 A5: Circle all of the transformations below that are linear. (a) T : R 3 R given by T (x 1, x, x 3 ) = (0, x, 3x 3 ). (b) T : R 3 R 3 given by T (x 1, x, x 3 ) = (x 1, x, x 3). (c) T : R 3 R 3 given by T (x 1, x, x 3 ) = (1, x, 3x 3 ). (d) T : R 3 R 3 given by T (x 1, x, x 3 ) = (x 1, x, 3x 3 ). A7: Compute the following product A8: Circle all the sets below which are subspaces of R 3? (a) The plane x + y + 3z = 0. (b) The intersection of planes x + y + 3z = 1 and x + y + z = 0. (c) The intersection of a plane x + y + 3z = 0 and the line through (0, 0, 0) and ( 1, 1, ). (d) The intersection of planes x + y + 3z = 1 and x + y + z = 1. 4

13 A9: Let A be an invertible n n matrix. Circle all true statements below. (a) The system of equations Ax = b is consistent for any b R n. (b) The column space of A is all of R n. (c) For any b R n, the set of vectors x R n such that Ax = b is a subspace of R n. (d) If AB = BA for a matrix B, then B is the inverse of A. A10: Circle all of the matrices below that are invertible? (a) (b) (c) (d) 1 M = M = M = M =

14 Part B - Long Answer Questions, 5 marks each. B1: Consider the system of linear equations below in variables x 1, x, x 3. x 1 x 3 = 1 x 1 + x = 1 x + x 3 = 1 (a) [1 mark] Write the augmented matrix of the system. (b) [ marks] Find reduced echelon form of the augmented matrix using row-reduction procedure. (c) [1 marks] Find the general solution of the system in parametric vector form. (d) [1 mark] Find the solution of the system with x 1 = 100. Show your work. 6

15 B: Let k be a real number and 1 0 k A = k (a) [1 mark] Show that A is invertible for any value of k. (b) [ marks] Find B = A 1. (Your answer will depend on k). (c) [ marks] Calculate (B 1 ) 3 A B 4. Show your work. 7

16 B3: Consider the linear transformations below. For each one write down its standard matrix. Show your work. a) [1 mark] T : R R rotates every vector by an angle of π about the origin and multiplies the resulting vector by scalar 4. b) [ marks] T : R 3 R 3 defined by T 0 = 1, T 0 =, T 1 = c) [ marks] Are the linear transformations above onto? Are they one-to-one? Explain. 8