Reduction Again - 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: Reduction Again, Echelon Form, Rank, Column Space, Basis, Row Space, Equation, Columns, Linear Combination, Solutions

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The University of British Columbia
Final Examinations April 25, 2010
Mathematics 221, Section 202
Instructor: V. Vatsal
Time: 2.5 hours
Name:
Student Number:
Signature:
Section Number:
Special instructions:
1. No books or notes or electronic devices allowed.
2. Answer all questions. Each part of each question is worth 2 marks, for a
total of 50 marks.
3. Give your answer in the space provided. If you need extra space, use the
back of the page.
4. Show enough of your work to justify your answer. Show ALL steps.
Rules governing examinations:
1. Each candidate must be prepared to produce, upon request, a UBCcard for identifica-
tion.
2. Candidates are not permitted to ask questions of the invigilators, except in cases of
supposed errors or ambiguities in examination questions.
3. No candidate shall be p ermitted 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.
4. 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:
(a) having at the place of writing any books, papers or memoranda, calculators, com-
puters, sound or image players/recorders/transmitters (including telephones), or
other memory aid devices, other than those authorized by the examiners; speak-
ing or communicating with other candidates; and
(b) purposely exp osing written papers to the view of other candidates or imaging
devices. The plea of accident or forgetfulness shall not be received.
5. 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. Candidates must follow any additional
examination rules or directions communicated by the instructor or invigilator.
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The University of British Columbia Final Examinations – April 25, 2010 Mathematics 221, Section 202 Instructor: V. Vatsal Time: 2.5 hours

Name: Student Number:

Signature: Section Number:

Special instructions:

  1. No books or notes or electronic devices allowed.
  2. Answer all questions. Each part of each question is worth 2 marks, for a total of 50 marks.
  3. Give your answer in the space provided. If you need extra space, use the back of the page.
  4. Show enough of your work to justify your answer. Show ALL steps.

Rules governing examinations:

  1. Each candidate must be prepared to produce, upon request, a UBCcard for identifica- tion.
  2. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
  3. 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.
  4. 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: (a) having at the place of writing any books, papers or memoranda, calculators, com- puters, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners; speak- ing or communicating with other candidates; and (b) purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received.
  5. 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. Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.

Problem 1: Let A =

 and let^ U^ =

Then the matrix U is an echelon form for A (you may assume this, you don’t have to do the row reduction again.)

a) Find the rank of A

b) Find a basis for the column space of A

e) Find a basis for the null space of A.

f) Let a 1 , a 2 ,... , a 5 denote the columns of A. Express a 5 as a linear combination of a 1 , a 2 , a 4.

g) Find the dimension of the null space of AT^.

h) Find all solutions to the system of equations:

x + 2y + 3z − 2 w = 4 2 x + 5y + 8z − w = 6 x + 4y + 7z + 5w = 2 x + 3y + 5z + w = 2

(Hint: what is the augmented matrix for the system?)

Problem 3: Let A =

a) Write the characteristic polynomial of A

b) Find all eigenvalues of A. (You don’t have to find the eigenvectors.)

Problem 4: Compute the matrix product

Problem 5: True or false (explain your answer): If A is a square matrix, then det(−A) = − det(A)

Problem 6: True or false (explain your answer): Suppose A is a 5 × 4 matrix and that b, c are vectors in R^5. Given that Ax = b has a unique solution, then Ax = c has a unique solution as well.

Problem 8: Suppose B = {v 1 , v 2 } is a basis for R^2 , and that C = {w 1 , w 2 , w 3 } is a basis for R^3. Suppose that T : R^2 → R^3 is a linear transformation such that T (v 1 ) = w 1 + w 2 and T (v 2 ) = w 1 + w 3. Find the matrix for T relative to the bases B and C.

Problem 9: Let W be the subspace of R^3 spanned by

 (^) and

. Find

the vector in W which is closest to v =

Problem 10: Let A =

. Then, given that A =

(you can assume this equation is true) find the matrix A^10. (There may be some powers of 2 in the answer, but you should get a single matrix.)

Problem 11: Find the determinant of the matrix