MATH 136 Final Exam
December 18, 2006
Time allowed: 150 minutes
Instructions
1. Attempt all problems.
2. The relative weights of problems are as shown below. The problems are in no particular order.
3. Be sure to show work, for otherwise it is impossible
Math 136
Assignment 3
Due: Wednesday, Jan 28th
3
2
1 and v = 1 . Calculate projv (u) and perpv (u).
1. Let u =
3
1
2. Find a scalar equation for the plane in R3 with vector equation
3
1
1
x = s 1 + t 1 + 1 , s, t R
3
1
2
2
2
3
3. Find the pr
Math 136
Assignment 9
Due: Wednesday, Mar 25th
1. Calculate the determinant of the following matrices.
1
2 4 0 0
7
1 2 2 9
(b) B =
(a) A =
2
3 6 0 3
3
1 1 0 0
10
7
2
3
7 9
7 7
6 2
4 1
a+b p+q u+v
a p u
2. Prove that det b + c q + r v + w = 2 det b q v
Math 136
Assignment 10
1. Calculate the area of the parallelogram determined by
Due: Wednesday, Apr 1st
3
1
,
4
2
1
2. Find the volume of the parallelepiped determined by 1,
4
1
2
3, 1
4
5
2
2
2 2 2
0
3. Let L be a linear mapping with [L] = 2 1 2
Math 136
Assignment 11
NOT HANDED IN
1. Determine the matrix of the linear operator L : R3 R3 with respect to the basis
B and determine [L(x)]B where B = cfw_v1 , v2 , v3 , L(v1 ) = 2v1 v2 3v3 ,
2
1.
L(v2 ) = 4v1 + 3v2 v3 , L(v3 ) = v2 , and [x]B =
1
2.
Math 136
Assignment 6 Solutions
1. Find a basis for the kernel, and a basis for the range of the linear mapping L : R3 R4
dened by
L(x1 , x2 , x3 ) = (x1 + 2x2 + 3x3 , x1 + x2 + x3 , x3 , x1 + x2 2x3 )
Solution: Every vector x Ker(L) satises
x1 + 2x2 + 3x
Math 136
1. For each of
invertible.
1
(a) A = 2
2
Assignment 8 Solutions
the following matrices, nd the inverse, or show that the matrix is not
2 4
2 3
3 5
Solution: Row reducing
1
2
2
[A | I] gives
1 0 0 1 2 2
2 4 1 0 0
2 3 0 1 0 0 1 0 4 13 11
3 5 0
Math 136
Assignment 7
Due: Wednesday, Mar 11th
1. Find a basis and the dimension of each of the following subspaces.
(a) S1 =
a b
|a+b=d
0 d
(b) S2 = Span
of M22 (R).
1 2
1 1
1 5
3 1
,
,
,
1 1
1 0
1 4
3 2
.
2. Given that B = cfw_1 + x + x2 , 1 + 3x + 2x2
Math 136
Assignment 8
Due: Wednesday, Mar 18th
1. For each of the following matrices, nd the inverse, or show that the matrix is not
invertible.
1 2 4
(a) A = 2 2 3
2 3 5
1 0 2
(b) B = 1 1 3
3 1 7
2. Find all right inverses of A =
2 1 5
.
3 2 1
1
2 3
3. W
Math 136
Assignment 4
Due: Wednesday, Feb 4th
1. For each of the following systems of linear equations:
i) Write the augmented matrix.
ii) Row-reduce the augmented matrix into RREF.
iii) Find the general solution of the system or explain why the system is
Math 136
Assignment 4 Solutions
1. For each of the following systems of linear equations:
i) Write the augmented matrix.
ii) Row-reduce the augmented matrix into RREF.
iii) Find the general solution of the system or explain why the system is inconsistent.
Math 136
Assignment 5
Due: Wednesday, Feb 25th
2 3
1 2 3
4 2 0
1. Let A =
,B=
, C = 1 1. Determine the following.
3 1 1
1 5 6
3 4
(a) 2A B
(b) (A + B)C
(c) C T B T + C T AT
2. Find t1 , t2 , t3 R such that
t1
3. Let A =
1 0
2 1
1 1
4 0
+ t2
+ t3
=
1 0
2 2
Math 136
Assignment 4 Solutions
1. For each of the following systems of linear equations:
i) Write the augmented matrix.
ii) Row-reduce the augmented matrix into RREF.
iii) Find the general solution of the system or explain why the system is inconsistent.
Math 136
Assignment 7 Solutions
1. Find a basis and the dimension of each of the following subspaces.
(a) S1 =
a b
|a+b=d
0 d
of M22 (R).
Solution: Every vector in S1 has the form
a b
a
b
1 0
0 1
=
=a
+b
0 d
0 a+b
0 1
0 1
1 0
0 1
,
spans S1 . Since neithe
Wednesday, January 8 Lecture 2 : Linear independence of vectors in n.
1. Linearly independent subset in n.
2. Characterization of a linearly independent set as one being a set where no vector
is a linear combination of the others.
3. A plane in 3. A hyper
MATH 136 - Assignment 7
Due Wednesday July 2 at 8:30am in the assignment dropbox
1. Determine which of the following sets is linearly independent.
(a) cfw_1 + 3x + x2 , x + x2 , 1 x P3 (R)
Solution: Consider the equation
0 = c1 (1 + 3x + x2 ) + c2 (x + x2
MATH 136 - Assignment 9
Due Wednesday July 16 at 8:30am in the assignment dropbox
1. For each of the following matrices, nd the inverse, or show that the matrix is not
invertible.
4 1
3 7
(a) A =
Solution: Row reduce [A | I]:
4 1 1 0
3 7 0 1
1 0 7/25 1/
Math 136
Assignment 10 Solutions
1. Calculate the area of the parallelogram determined by
Solution: A = det
3 1
4 2
2. Find the volume of the
1 1
Solution: V = det 1 3
4 4
3
1
,
4
2
= |3(2) (1)(4)| = 10
1
parallelepiped determined by 1,
4
2
1 1 2
1 = de
MATH 136 Summer 2014 Section 001
Course Info
Lecture:
MC 4061
MWF 2:30-3:20pm
Tutorial:
MC 2017
M 4:30-5:20pm
Instructor
Patrick Roh
MC 6494
[email protected]
Office Hours
No appointment needed:
MWF 1-2pm and 5:30pm (will continue as long as students
MATH 136 (Spring 2014) Section 002
Section information
Lectures:
MWF 3:304:20pm
MC 4061
Tutorial:
T 2:303:20pm (starting the second week of class)
MC 2017
Instructor
Darryl Hoving
[email protected] (include MATH 136 in the subject)
Office: QNC 3321 (N