MATH 1300 University of Manitoba Linear Algebra Final Exam

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UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
Name:
Student Number:
I understand that cheating is a serious offence:
(Signature – In Ink )
INSTRUCTIONS
I. No calculators, texts, notes, cell phones, pagers, translators or other electronics are
permitted. No outside paper is permitted.
II. This exam has a title page, 26 pages including this cover page and 1 scrap page for
rough work. Please check that you have all the pages.
III. The value of each question is indicated in the lefthand margin beside the statement of
the question. The total value of all questions is 85 points.
IV. Answer all questions on the exam paper in the space provided beneath the question.
If you need more room, you may continue your work on the reverse side of the page,
but CLEARLY INDICATE that your work is continued.
V. Work on pages without questions will only be marked if it is clearly linked to a question
(e.g., “Question 3 (continued)”).
VI. Do not make any marks on the QR codes, that would probably lead to the page not
being recognized (and as a consequence, not being marked). Do not write too close to
the staple, as the area near it will be chopped off for scanning.
VII. Show all your work clearly and justify your answers using complete sentences. Unjustified answers will receive LITTLE or NO CREDIT.
VIII. If a question calls for a specific method, no credit will be given for any other method.
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
1. Suppose that the augmented matrix of a system of linear equations has been partially
reduced using elementary row operations to


1 1 3 | 5
 0 1 1 | a .
0 b 1 | 2
[5]
(a) Find all values (if any) of a and b for which the system is inconsistent.
[3]
(b) Find all values (if any) of a and b for which the system has a unique solution.
[2]
(c) Find all values (if any) of a and b for which the system has infinitely many solutions.
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours


2 2 −3
2 . Given that |BA + B| = 10, find AT B .
[6] 2. Let A =  0 3
0 0
4
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
3. Let P (−1, 1, 1), Q(1, 2, 3) and R(2, 1, 0) be points in R3 .
(a) Find an equation of the plane in R3 containing P , Q and R. Write the equation in
the form ax + by + cz + d = 0.
[6]
[4]
(b) Find parametric equations of the line L in R3 passing through P and Q.
⇒ Continued on the next page
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
[4]
(c) Find the point of intersection of L, found in part (b), and the xy-plane.
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
[7] 4. Fill in the blanks. No justification required.
If A is an n × n matrix, then the following statements are equivalent.
1. A is invertible.
2. Ax = 0 has only
.
.
3. The reduced row echelon form of A is
.
4. A is expressible as a product of
for every n × 1 matrix b.
5. Ax = b is
for every n × 1 matrix b.
6. Ax = b has exactly
7. det(A) 6=
8.
.
is not an eigenvalue of A.
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
[4] 5. (a) Find projv u, the projection of u along v, where u = (2, −2, 3) and
v = (2, 1, 3).
[3]
(b) Find the value of t so that the vector (2, 5, −3, 6) is orthogonal to (4, t, 7, 1).
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
6. Consider the transformation T : R3 → R3 defined by
T (x, y, z) = (−y, 2x + y, x).
[6]
(a) Prove, by using the definition, that T is a linear transformation.
[3]
(b) Find the standard matrix of the linear transformation T .
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
7. Consider the linear one-to-one transformation T : R2 → R2 defined by
T (1, 0) = (0, 1),
T (0, 1) = (−1, 0).
[3]
(a) Find the image of v = (1, 1) by T .
[2]
(b) What is the effect of T on any vector u = (x, y) in R2 ?
[3]
(c) Find the inverse of the standard matrix of the transformation T .
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours


2 0 −2 1 −1
0 −2 1
3
9


0
6
6
8. Let A = 
0 0
.
0 0
0 −1 1 
0 0
0
0 −1
[4]
(a) Find the eigenvalues of A.
[2]
(b) Find A−1 if possible. Justify.
[6]
(c) Find the eigenvectors associated to the eigenvalue −1.
⇒ Continued on the next page
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
[2]
(d) Is A diagonalizable? Justify.
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours


2 −2
9. Let A =
. The eigenvalues of A are λ1 = 4 and λ2 = −1.
−3 1
[2]
(a) Give the characteristic polynomial of A.
[8]
(b) Find a diagonal matrix D and an invertible matrix P such that D = P −1 AP (There
is no need to find P −1 ).
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
DATE & TIME: April 12, 2018, 6:00–8:00
FINAL EXAMINATION
DURATION: 2 hours
[Scrap Page]
UNIVERSITY OF MANITOBA
COURSE: MATH 1300
FINAL EXAMINATION
DATE & TIME: April 12, 2018, 6:00–8:00
DURATION: 2 hours