December 2005 Mathematics 307 Page 2 of 9 pages
Marks
[10] 1.
(a) [5] Compute the LU decomposition of the matrix A=
1 1 1 0
2 3 4 1
0−111
(b) [5] If xsolves the equation Ax=
1
0
1
, find Ux.
Continued on page 3
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December 2005 Mathematics 307 Exam: Linear Algebra and Differential Equations, Exams of Linear Algebra
The december 2005 exam for mathematics 307: applied linear algebra at the university of british columbia. The exam covers various topics in linear algebra, including matrix decompositions, eigenvalues and eigenvectors, orthogonal projections, and differential equations. Students are required to compute determinants, find eigenvectors and eigenvalues, perform matrix multiplication, and analyze stability of differential equations.
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2012/2013
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Marks
[10] 1.
(a) [5] Compute the LU decomposition of the matrix A =
(b) [5] If x solves the equation Ax =
, find U x.
[10] 2.
(a) [5] Compute the rank of
. What are the dimensions of the nullspace, the
column space, the row space, and the left nullspace?
(b) [5] Find an orthogonal basis for the space spanned by
in^ R^4.
[15] 4.
(a) [10] Show that eA+B^6 = eAeB^ for A =
[
]
and B =
[
]
. (Hint:
[
]
1 2
[
] [
] [
]
(b) [5] Let A and B be any two n × n matrices. Show that if λ is an eigenvalue of AB and λ 6 = 0 then λ is also an eigenvalue of BA.
[12] 5. Let X be the subspace in three dimensional space R^3 containing all vectors perpendicular to
. Let L be the linear transformation defined by first projecting a vector onto the x–y
plane and then projecting the resulting vector onto X.
(a) [4] Show that the vectors v 1 =
(^) and v 2 =
(^) form a basis for X.
(b) [4] Find the 3 × 3 matrix the represents L as a linear transformation from R^3 to R^3
(c) [4] If x lies in X then so does L(x). This means that L defines a linear transformation from X to X. Find the matrix of this transformation with repsect to the basis in part (a).
[10] 7. The equation (^)
[
x 1 x 2
]
has no solution.
(a) [7] Find the “least squares” solution.
(b) [3] Explain in what way it is the best possible approximation to a solution.
[12] 8.
(a) [4] Find the 3 by 3 matrix A with eigensystem λ 1 = 1, λ 2 = 2/ 3 , λ 3 = 0 and v 1 =
(^) , v 2 =
(^) , v 3 =
(b) [4] Calculate A^100
, up to negligible error.
(c) [4] What is the entry in the second row and second column of A^100?
The End