KOÇ UNIVERSITY FALL 2022
MATH 107, SAMPLE FIRST MIDTERM EXAM
Duration of Exam: 90 minutes
You can NOT use calculators in the exam.
No books, no notes, no questions and no talking allowed.
You must always explain your answers and show your work to receive full credit.
Print (use CAPITAL LETTERS) and sign your name below.
Name: —————————————————
Surname: —————————————————
Signature: ————————————————————
PROBLEM
1
2
3
4
5
TOTAL
POINTS SCORE
20
20
20
20
20
100
1
2
PROBLEM 1 (20 points): Consider a system of linear equations Bx = c, where c 2 R3 and
2
3
2 3
1
3
1 5 5
B=4 0 0
0 0
0 3
⇥
⇤T
Suppose that the vector a = 0 ⇡ 0 1
is a solution of the system Bx = c.
(a) (5 points) Find the vector c.
(b) (15 points) Write down the general solution of the system Bx = c.
3
PROBLEM 2 (20 points): Consider the matrix
2
1
C=4 2
2
(a) (10 points) Find all the values of k in R such
3
0 k
1 0 5
1 1
that the matrix C is noninvertible.
1
and let T : R3 ! R3 be the linear transformation given by T (x) = Cx.
2 2
3
1
Find T 1 (d) for the vector d = 4 0 5 2 R3 .
7
(b) (10 points) Take k =
4
PROBLEM 3 (20 points) : Consider the vectors
2
3
2
3
2
1
0
u=4 1 5 v=4 1 5 w=4
2
2
3
2
5 5
2
2
3
s
9 5 does NOT belong to
(a) (10 points) Find all the values of s 2 R such that vector p = 4
s+2
Span {u, v, w}.
(b) (10 points) Let G be the matrix whose columns are the vectors u, v, w. Consider the
2 linear
3
0
transformation S : R3 ! R3 defined as S(x) = Gx. Is there a vector q 2 R3 such that S(q) = 4 9 5?
2
Explain your answer.
5
PROBLEM 4 (20 points) : Consider the linear transformation T : R2 ! R4 given by
2
3
x1 + 7x2

6 4x1 + 5x2 7
x1
7.
T(
)=6
4
5
x2
3x2
5x1 + x2


1
0
(a) (5 points) Compute T (
) and T (
)
0
1
(b) (5 points) Write the standard matrix for the linear transformation T .
(c) (5 points) Is T onto? Give a reason for your answer.
(d) (5 points) Is T one-to-one? Give a reason for your answer.
6
PROBLEM 5 (20 points) : If the following statement is true, give a justification. If it is false, construct
a specific example to show that the statement is not always true.
If {v1 , v2 , v3 , v4 } is a linearly independent set of vectors in R4 , then {3v3 , v2 , 2v1
linearly independent.
v4 , 5v1 } is also