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THE UNIVERSITY OF MANITOBA
DEPARTMENT OF MATHEMATICS
I understand that cheating is a serious offense.
136.130 Vector Geometry and
Linear Algebra
Mid-Term Exam
Signature:
Date: Friday, Oct
1
136.130: Test #4
20 minutes
B16.
Solutions
1. Suppose u = (1, 0, 2) , v = (1,1, 1) , w = (0,1, 2) . Compute if possible, otherwise state it is not
possible.
(a) ui(v w)
(b) (uiv) w
(c) u (v (0, 0, 0
THE UNIVERSITY OF MANITOBA
DATE: October 25, 2004
Midterm Examination
DEPARTMENT & COURSE NO. 136.130
PAGE NO: 1 of 6
EXAMINATION: Vector Geometry & Linear Algebra
TIME: 1 Hour
Midterm exam, 136.130,
1
136.130: Test #5
Solutions
B13.
[7]
1. Which of the following is a subspace of the vector space M 2,2 of all 2 2 matrices (with
the usual addition and scalar multiplication)? Do NOT justify your ans
1
136.130: Test #3
Solutions
B15.
1. Solve the following system using Cramers rule. No points will be awarded if other methods are
2x + y = 5
used.
x y = 1
2 1
5 1
2 5
det(A1 ) 6
Solution. With A =
THE UNIVERSITY OF MANITOBA
DATE: October 28, 2002
MIDTERM EXAMINATION
PAGE 1 of 6
DEPARTMENT & COURSE NO: 136.130
TIME: 1 hour
EXAMINATION: Vector Geometry & Linear Algebra
EXAMINERS: Various
_
Values
136.130: Test #1 Solutions
B01.
1. Which of the following matrices are in Reduced Row-Echelon Form, which are in RowEchelon Form, and which are neither in RREF nor in REF? (No need to justify your ans
2. 136.130: Test #2
20 minutes
Name:_ Student Number: _
1. Suppose
and
. Find
(a)
(b)
2. Which of the following matrices are in Row Reduced Echelon form, which are
in Echelon form (but not in RRE form
1
136.130: Test #4 Solutions
4.
1. Find the rank of the following matrix.
- 1 0
(a) - 1 0 - 1 0 -
2
2
2
- 1 0 - 2
This matrix can obviously be row reduced to 0 0 0 so that the row reduced
0 0 0
ec
136.130: Test #2 Solutions
B02.
1. Find the inverse of each of the following two matrices, or show it does not exist.
,
.
Solution.
For A:
A is
after row reducing becomes
; so the inverse of
.
For B:
3. 136.130: Test #3
20 minutes
Name:_ Student Number: _
1. Suppose
and
. Compute the following if
possible, or state the reason why the operation could not be performed.
(a)
(b)
(c)
(d)
can not be don
B03.
136.130: Test #3 Solutions
1. Suppose
Solution.
. Compute
and
.
2. Suppose A and B and both 4 by 4 matrices and suppose
Find
and
and
.
.
Solution.
3.
(a) Use cofactors to compute the determinant
136.130: Test #4 Solutions
B04.
1. Suppose
and
. Find
(a)
(b)
Solution.
(a)
(b)
.
2. We are given a point
and a vector
.
(a) Find the point normal equation of the plane through P and orthogonal
to u.
1. 136.130: Test #5 Solutions
20 minutes
1. Can Cramers rule be used to solve the system
x only.
Solution. The coefficient matrix is
compute
Then
? If yes, find
. We
. So Cramers rule can be applied.
1
136.130: Test #1
Solutions
B13.
1. Use Gauss-Jordan elimination to solve the following system. Show your work
describing your steps. State clearly your final answer.
x +
y z = 0
2y 2z = 2
y + z = 1