221Midterm1 solutions

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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:
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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
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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
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(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
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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
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(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.
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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
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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.
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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
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(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

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


. 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
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
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Winter 2, 2013
Math 221 Midterm 1
Name:
Page 7 of 7 pages
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