UBC Math 307 Final Exam December 2008, Exams of Linear Algebra

The final exam for math 307 at the university of british columbia, december 2008. The exam consists of 10 problems covering various topics in linear algebra, including matrix rank, null spaces, qr decomposition, determinants, and singular values. Students are allowed one page of notes and must solve the problems without the use of textbooks or calculators. The exam lasts for 150 minutes.

Typology: Exams

2012/2013

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Math 307 Final Exam
Dec 3, 2008
Duration: 150 minutes
Last Name: First Name: Student Number:
Do not open this test until instructed to do so! This exam should have 12 pages,
including this cover sheet. No textbooks, calculators, or other aids are allowed. One page
of notes is allowed. Turn off any call phones, pagers, etc. that could make noise during
the exam. You must remain in this room until you have finished the exam. Circle your
solutions! Reduce your answer as much as possible. Explain your work. Relax.
Use the back of the page if necessary.
Read these UBC rules governing examinations:
(i) Each candidate must be prepared to produce, upon request, a Library/AMS card for identification.
(ii) Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors
or ambiguities in examination questions.
(iii) No 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.
(iv) Candidates suspected of any of the following, or similar, dishonest practices shall be immediately
dismissed from the examination and shall be liable to disciplinary action.
Having at the place of writing any books, papers or memoranda, calculators, computers, audio
or video cassette players or other memory aid devices, other than those authorized by the
examiners.
Speaking or communicating with other candidates.
Purposely exposing written papers to the view of other candidates. The plea of accident or
forgetfulness shall not be received.
(v) Candidates must not destroy or mutilate any examination material; must hand in all examination
papers; and must not take any examination material from the examination room without permission
of the invigilator.
Problem Out of Score
1 10
2 10
3 10
4 10
5 10
6 10
7 10
8 10
9 10
10 10
Total 100
1
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pf4
pf5
pf8
pf9
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Math 307 Final Exam Dec 3, 2008 Duration: 150 minutes

Last Name: First Name: Student Number: Do not open this test until instructed to do so! This exam should have 12 pages, including this cover sheet. No textbooks, calculators, or other aids are allowed. One page of notes is allowed. Turn off any call phones, pagers, etc. that could make noise during the exam. You must remain in this room until you have finished the exam. Circle your solutions! Reduce your answer as much as possible. Explain your work. Relax. Use the back of the page if necessary. Read these UBC rules governing examinations:

(i) Each candidate must be prepared to produce, upon request, a Library/AMS card for identification. (ii) Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. (iii) No 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. (iv) Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action.

  • Having at the place of writing any books, papers or memoranda, calculators, computers, audio or video cassette players or other memory aid devices, other than those authorized by the examiners.
  • Speaking or communicating with other candidates.
  • Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received. (v) Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator.

Problem Out of Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100

Let

A =

Determine the rank of A and find a basis of the left null space N (A>) of A.

For the parameter α, consider the linear system:

x − y + z = 2 , − 2 x + y + αz = − 3 , x + αy − z = 1.

Determine when the system has a unique solution, no solution, or infinitely many solutions. (Don’t determine the actual solutions!)

Find the QR decomposition of the matrix

A =

Let u be a unit vector and Q = I − 2 uu>. Show:

(a) The matrix Q is symmetric and orthogonal.

(b) The matrix Q + iI is invertible.

Consider the symmetric matrix

A =

Compute the matrix norms ‖A‖∞ and ‖A‖ 2.

Find the singular values of

A =

Then determine the diagonal matrix Σ in the singular value decomposition A = U ΣV >.

Let AH^ = −A be a skew-Hermitian complex matrix. Show that the matrix eAt^ is unitary.