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The University of British Columbia First Midterm Examination Mathematics 221 Matrix Algebra Closed book examination Last Name Time: 50 minutes First Signature Student Number Special Instructions: sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m No calculators, open books, or notes are allowed. There are 5 problems in this exam. Rules governing examinations Th • Each examination candidate must be prepared to produce, upon the request of the invigilator or examiner, his or her UBCcard for identification. • Examination candidates are not permitted to ask questions of the examiners or invigilators, except in cases of supposed errors or ambiguities in examination questions, illegible or missing material, or the like. • No examination candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination. Should the examination run forty-five (45) minutes or less, no examination candidate shall be permitted to enter the examination room once the examination has begun. • Examination candidates must conduct themselves honestly and in accordance with established rules for a given examination, which will be articulated by the examiner or invigilator prior to the examination commencing. Should dishonest behaviour be observed by the examiner(s) or invigilator(s), pleas of accident or forgetfulness shall not be received. • Examination candidates suspected of any of the following, or any other similar practices, may be immediately dismissed from the examination by the examiner/invigilator, and may be subject to disciplinary action: i. speaking or communicating with other examination candidates, unless otherwise authorized; ii. purposely exposing written papers to the view of other examination candidates or imaging devices; iii. purposely viewing the written papers of other examination candidates; iv. using or having visible at the place of writing any books, papers or other memory aid devices other than those authorized by the examiner(s); and, v. using or operating electronic devices including but not limited to telephones, calculators, computers, or similar devices other than those authorized by the examiner(s)(electronic devices other than those authorized by the examiner(s) must be completely powered down if present at the place of writing). • Examination candidates must not destroy or damage any examination material, must hand in all examination papers, and must not take any examination material from the examination room without permission of the examiner or invigilator. • Notwithstanding the above, for any mode of examination that does not fall into the traditional, paper-based method, examination candidates shall adhere to any special rules for conduct as established and articulated by the examiner. • Examination candidates must follow any additional examination rules or directions communicated by the examiner(s) or invigilator(s). Page 1 of 7 pages https://www.coursehero.com/file/9307238/221Midterm1-solutions/ 1 8 2 10 3 10 4 10 5 12 Total 50 Winter 2, 2013 Math 221 Midterm 1 Name: Page 2 of 7 pages PROBLEM 1. For each of the following, circle “T” is the statement is true, and “F” if the statement if false. (i) The equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row. T F sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m (ii) A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution. T F (iii) Rm is equal to the span of the columns of an m × n matrix A if and only if A has a pivot position in every column. T F (iv) The equation Ax = b is homogeneous if the zero vector is a solution. T F (v) A homogeneous system of linear equations can be inconsistent. T F (vi) If p1 and p2 are solutions to the linear system Ax = b, then p1 − p2 is a solution to Ax = 0. F Th T (vii) If the set {u, v, w} is linearly independent, then the set {u − v, v − w, w − u} is linearly independent. T F (viii) If {v1 , v2 , v3 , v4 } is a linearly independent set, then {v1 , v2 , v3 } is also linearly independent. T https://www.coursehero.com/file/9307238/221Midterm1-solutions/ F Winter 2, 2013 Math 221 Midterm 1 Name: Page 3 of 7 pages PROBLEM 2. A box of cereal typically lists the number of calories, and the amounts of protein, carbohydrate, and fat contained in one serving of the cereal. The amounts for cereals A and B are given below. Suppose that you wish for a mixture of these two cereals that contains exactly 295 calories, 9 grams of protein, 48 grams of carbohydrate, and 8 grams of fat. Nutrient Calories Protein (grams) Carbohydrate (grams) Fat (grams) Cereal A 110 4 20 2 Cereal B 130 3 18 5 sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m (a) Set up a vector equation for this problem. Include a statement about what variables in your equation represent. Th (b) Write an equivalent matrix equation. Determine if the desired mixture of the two cereals can be prepared, and if so, give the required proportions of each cereal. https://www.coursehero.com/file/9307238/221Midterm1-solutions/ Winter 2, 2013 Math 221 Midterm 1 Name: Page 4 of 7 pages PROBLEM 3. Find the general solution of 3x1 − x2 + x3 + 2x4 = 16 x1 − x2 + x3 + 2x4 = 6 2x1 − 2x2 + 2x3 + kx4 = 13 Th sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m where k is an arbitrary constant. Express the solution in the parametric vector form. Explain how diﬀerent values of k aﬀect the set of solutions. https://www.coursehero.com/file/9307238/221Midterm1-solutions/ Winter 2, 2013 Math 221 Midterm 1 Name: PROBLEM 4. Suppose that a linear system has a form REF (A) is 1 −2 0 0 REF (A) = 0 0 0 0 Page 5 of 7 pages coeﬃcient matrix A whose reduced echelon 0 1 0 1 −2 0 0 0 1 0 0 0 sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m (a) Express the solution set for the homogeneous linear system Ax = 0 in parametric vector form. 1 1 2 2 (b) Suppose that a solution to the matrix equation Ax = 3 is the vector 3 −2 4 −1 another solution to this matrix equation. Th https://www.coursehero.com/file/9307238/221Midterm1-solutions/ . Find Winter 2, 2013 Math 221 Midterm 1 Name: Page 6 of 7 pages Problem 5. For each matrix below, determine whether its columns span R3 . Then determine whether its columns are linearly independent. If they are linearly dependent, provide a non-trivial linear relation between the columns. −4 −2 5 5 0 (a) 2 6 7 −3 1 −3 (b) −4 12 −3 8 1 1 −5 3 (c) 0 2 −3 2 −2 4 1 0 Th sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m https://www.coursehero.com/file/9307238/221Midterm1-solutions/ Winter 2, 2013 Math 221 Midterm 1 Name: Page 7 of 7 pages Th sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m Empty Page. https://www.coursehero.com/file/9307238/221Midterm1-solutions/ Powered by TCPDF (www.tcpdf.org) The End