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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
Typology: Exams
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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:
Rules governing examinations:
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