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Math 111 Midterm Exam
March 25, 2004
Your Name
Student Number
Section Number
1.
You may look at your book and lecture notes. For more space, write on the opposite side.
2.
Show all your work. Cross o. (instead of erase) the undesired part.
3.
Provide all the details. Your reasonings count most of the points.
Number Score
1
2
3
4
5
Total
1
201 0
110 .1
1
A =
,b =
.
..
..
011 .2
1
1) Solve the homogeneous equation Ax = 0 and express the solution set in terms of span of some
vectors in R4 . 2) Solve the equation Ax = b. 3) Find the rank of A.
Solution.
..
..
..
..
..
..
..
..
1
.2
1
.2
201 0 1
00 3
00 3
.1
.1
10
.1
10
.1
110
1
10
10
[Ab]=
.2
.2
.2
011
1
011
1
01 1
1
01 1
1
so
..
..
..
100 1/31/3
.2
.4/3
01 1
1
010
2/3
[Ab]=
.2
.2/3
00 3
1
001
1/3
1) Let x4 = r, then x1 = .r/3, x2 =4r/3 and x3 =2r/3. So the solution set of equation Ax =0
is the span o
. .
.
.1/3
4/3
2/3
1
. .
.
.
2) The general solution of equation Ax = b is
x = r
. .
.
. .
.
. .
.
. .
.
.1/3
4/3
2/3
1/3
2/3
1/3
+
1
0
where r is a free parameter.
3) The rank of A is three because there 3 pivotal columns.
T (x1,x2,x3)=(x1 +2x2, 2x1 +3x2, 3x1 +4x2).
1) Find the standard matrix A for T .
2) Is T onto?
3) Is T one-to-one?
Please provide reason. Solution.
A =[T (e1),T (e2),T (e3)] =
..
..
.
340
The last column of A is a non-pivotal column, so T is not one-to-one and not onto.
1121
0113
h
A =
.
,b =
0011
0
0011
h
1) Does Ax = 0 have nontrivial solution? 2) For what h, does Ax = b have a solution? 3) What are det A, det(5A) and det(A5)? 4) Are columns of A linearly independent? 5) Do columns of A span R4?
Please provide reason. Solution.[A, b] is row equivalent to
. .
.
00110
0000 h
. .
.
.
It is then clear that A has three pivotal columns and one non-pivotal column and equation Ax = b has a solution if and only if h = 0. So 1) Equation Ax = 0 has nontrivial solutions; 2) h = 0; 3) det A = 0, det(5A)=54 det A = 0 and det(A5) = (det A)5 = 0; 4) The columns of A are not linearly independent; 5) The columns of A does not span R4 .
4 (21 points) 1) Find two 2 2 matrices A and B such that A, B are not zero, but AB = 0. 2) Find two matrices A and B such that AB is an identity matrix, but A and B are not invertible. 3) Find a matrix A such that the rank of A, A2 are 1 and 0 respectively.
Solution. 1) Take .01.
A = B = .
00
2) Take A = [1, 0] and B be the transpose of A, then AB = [1].
3) Take
.01.
A = .
00 Then A2 is a zero matrix. So the rank of A2 is zero. It is clear that the rank of A is 1.
in 1 . 5, v1,v2,v3,v4,v5 are vectors in R3
1) If v1,v2,v3,v4,v5 span R3 then thr