Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Exam of Matrix Algebra which includes Eigenvalues, Electronic Aids, System, Values, Parametric Form, Invertible, Non Zero Eigenvector, Matrix etc.Key important points are: Column Space, Denote, Matrix, Vector, Space, Invertible, Eigenvalue, Determinant, Calculation, Plane
Typology: Exams
1 / 14
This page cannot be seen from the preview
Don't miss anything!
Math 221: Final Exam Date: December 6, 2008
Name (please print):
Student Number:
Section number:
Instructions: No textbook, notes, or other aids allowed. Show all your work. If you need more space, use the back of the page. Each problem is worth 10 marks (5+5).
There are 14 pages in this exam. The last 3 pages are blank for scratch work. Please return all 14 pages.
Problem 1a: [5] Let A denote the matrix A =
. Find a basis
for the column space of A.
Problem 1b: [5] Determine whether the vector b =
(^) is in the column
space of the matrix A. above.
Problem 3: [5] Show that the determinant of the matrix A =
1 x x^2 1 y y^2 1 z z^2
is given by (z − x)(y − x)(z − y).
Problem 3b: [5] Find the determinant of the matrix A =
(Hint: you can use part (a) above to save some calculation.)
Problem 4a: [5] Let W be the plane in R^4 spanned by the vectors v =
and w =
. Verify that the vector^ u^ =
is in the subspace^ W^ and show
that the vectors u and w form an orthogonal basis for W.
Problem 4b: [5] Find the vector in W which is closest to b =
Problem 6a: [5] Suppose that each year 10% people living in Alberta move to BC, while 15% of people living in BC move to Alberta. Write a transition matrix that represents the change in population each year.
Problem 6b: [5] Find limiting population distribution if in the initial year, there at 100,000 people living in each province BC and Alberta.
Problem 7a: [5] Find the characteristic polynomial of the matrix
Give your answer in the form −λ^3 + aλ^2 + bλ + c where a, b, c are real numbers.
Problem 7b: [5] True or false (explain your answer): If A and B are similar matrices, then A and B have the same eigenvalues.
Problem 9a: [5] Let B = {b 1 , b 2 } be a basis for R^2 and let T be the linear transformation R^2 → R^2 such that T (b 1 ) = 2b 1 + b 2 and T (b 2 ) = b 2. Find the matrix of T relative to the basis B.
Problem 9b: [5] Suppose now that b 1 =
and b 2 =
. Find the matrix
of T relative to the standard basis of R^2.
Problem 10a: [5] True or false (explain your answer): Suppose that v 1 ,... , vn are a basis for Rn, and A is an invertible n × n matrix. Then the vectors Av 1 ,... , Avn are also a basis for Rn.
Problem 10b: [5] True or false (explain your answer): if A is a 2 × 2 matrix with characteristic polynomial (λ − 2)^2 then it is diagonalizable.
