Linear System - Matrix Algebra - Exam, Exams of Algebra

This is the Exam of Matrix Algebra which includes Unique Solution, System, Solutions, Infinitely Many Solutions, General Solution, Parametric Vector, Determinants, Matrices, Traffic Flow etc. Key important points are: Linear System, Values, Solution, System, Map, Rotates Points, Oriented, Formula, Linear Transformation, Standard Matrix

Typology: Exams

2012/2013

Uploaded on 02/25/2013

ekanath
ekanath 🇮🇳

3.8

(4)

76 documents

1 / 12

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
The University of British Columbia
Final Examination - April 20, 2007
Mathematics 221
Sections 201, 202, 203
Instructors: Dr. Macasieb, Dr. Tsai, and Dr. Liu
Closed book examination Time: 2.5 hours
Name Signature
Student Number
Special Instructions:
- Be sure that this examination has 12 pages. Write your name on top of each page.
- No calculators or notes are permitted.
- Show all your work. Unsupported solutions deserve no mark.
- In case of an exam disruption such as a fire alarm, leave the exam papers in the room and
exit quickly and quietly to a pre-designated location.
Rules governing examinations
Each candidate should be prepared to produce her/his
library/AMS card upon request.
No candidate shall be permitted to enter the examination
room after the expiration of one half hour, or to leave during
the first half hour of examination.
Candidates are not permitted to ask questions of the in-
vigilators, except in cases of supposed errors or ambiguities
in examination questions.
CAUTION - Candidates guilty of any of the following or
similar practices shall be immediately dismissed from the
examination and shall be liable to disciplinary action.
(a) Making use of any books, papers, or memoranda, other
than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other
candidates.
Smoking is not permitted during examinations.
1 12
2 10
3 10
4 12
5 10
6 12
7 7
8 12
9 15
Total 100
Page 1 out of 12
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Linear System - Matrix Algebra - Exam and more Exams Algebra in PDF only on Docsity!

The University of British Columbia Final Examination - April 20, 2007 Mathematics 221 Sections 201, 202, 203 Instructors: Dr. Macasieb, Dr. Tsai, and Dr. Liu

Closed book examination Time: 2.5 hours

Name Signature

Student Number

Special Instructions:

  • Be sure that this examination has 12 pages. Write your name on top of each page.
  • No calculators or notes are permitted.
  • Show all your work. Unsupported solutions deserve no mark.
  • In case of an exam disruption such as a fire alarm, leave the exam papers in the room and exit quickly and quietly to a pre-designated location.

Rules governing examinations

  • Each candidate should be prepared to produce her/his library/AMS card upon request.
  • No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of examination.
  • Candidates are not permitted to ask questions of the in- vigilators, except in cases of supposed errors or ambiguities in examination questions.
  • CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers, or memoranda, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates.
  • Smoking is not permitted during examinations.

Total 100

Page 1 out of 12

  1. [12pt] Consider the following linear system

x + 3y − 2 z + 2w = 1 y + z − 2 w = 2 x + 2y − 2 z + aw = 0 2 x + 8y − z + w = b

For which values of a and b, if any, does the system have: (Justify you answers!!) (i) No solution? (ii) Exactly one solution? (iii) Exactly two solutions? (iv) More than two solutions?

  1. [10pt] For what values of k is the matrix A =

0 1 k

 (^) invertible? When it is invertible,

find its inverse.

  1. [12pt] Let W =

b + 2c − d 2 b + 4c − d d −b − 2 c + d

b, c, d real

(i) Find a matrix A such that Col A = W.

(ii) Find a basis for W.

(iii) If B =

0 2 0 k 1 1 1 3

 and dim (Row^ B) = 2, find the value of the constant^ k.

  1. [10pt] Let A =

x 1 1 1 1 1 x 1 1 1 1 1 x 1 1 1 1 1 x 1 1 1 1 1 x

. Find all values of x such that A is not invertible.

  1. [12pt] Let P 2 be the vector space of polynomials of degree at most 2.

(i) The set B = {1 + t, 1 + t^2 , t + t^2 } is a basis for P 2. Find the coordinate vector [2+t−t^2 ]B.

(ii) The set C = {1 + t^2 , t + t^2 , 1 + t} is also a basis for P 2. Find ~p(t) in P 2 such that ~p(1) = 1 and [~p(t)]B = [~p(t)]C.

  1. [12pt] Suppose

w ~ 1 =

 (^) , ~w 2 =

 (^) , ~w 3 =

 (^) , ~y =

Let W = Span{ w~ 1 , ~w 2 , ~w 3 }. (i) Determine the dimension of W and find a basis for W. (ii) Find an orthogonal basis for W , and the orthogonal projection of ~y onto W. (iii) What is the shortest distance from ~y to W?

  1. [8/2/5pt] The matrix M =

(i) Verify that M has eigenvalues 0 and 3, and find the corresponding eigenspaces. (ii) What is the rank of M? (iii) Is M diagonalizable? Is there an orthogonal set of eigenvectors of M that forms a basis of R^3? Justify your answers.