LA with DE Midterm Exam

MATH 120 24701 Spring, 2024

Honor Statement:
By signing below you confirm that you have neither given nor received any unauthorized
assistance on this test. Furthermore, you agree not to discuss this test with anyone until
the testing period is over. Sign here

Name:

Roll Number:

Instructor’s Name:

Exam Hall & Seat Number:

Time Allowed: 120 minutes Total Marks: 40

DO NOT OPEN THIS EXAM UNTIL TOLD TO DO SO.

Instructions:

• It is a closed book/closed notes exam.

• Calculators are NOT allowed.

• Blue books (Answer sheets) are for rough work only. Blue books will not be marked
or graded.

• Correct options for the MCQs should be filled out/written carefully. No

overwriting/rewriting is allowed.

• Please hand over the white answer sheet and the blue books separately to the
invigilators once you complete your exam.

MCQs marked anywhere else will not be checked. No overwriting of MCQs is

permitted on the MCQ Answer sheet. MCQs marked with lead pencils will not be checked.

Questions : MCQ’s 1 2 3 4 5 6 Total

Score :

Lahore University of Management Sciences, Mar 09, 2024

DHA, Lahore 54792, Pakistan
http://www.lums.edu.pk
Each of the following questions, or incomplete statements, is followed by
suggested answers, or completions. In each case, select the one that is the best
of the choices.

1. If u, v are solutions of a linear system Ax = b, then 2u + 5v is a solution of Ax =

(A) b (B) 0 (C) 3b (D) 3b (E) None of the other

2. Consider the vectors u = (1, 2, 3, 4, ..., 2024), v = (1, 1, 1, 1, ..., 1, 1) in R2024 , then
the angle between them is
(A) acute (B) obtuse (C) right (D) angle is not defined in R2024

3. The
✓ distance◆ between
✓ ◆the point P (1, 2, 5) and the plane passing through the points
7 7
A , 0, 0 , B 0, , 0 , C(0, 0, 7) is
3 4
p
9 26 5 3 16
(A) p (B) p (C) p (D) p
26 26 26 26
(E) None of the other choices is correct

2 3
1 3 1 4
60 5 7 27
4. Consider the matrix 6
40
7, then
0 0 55
0 0 0 6

a14 M14 + a24 M24 a34 M34 + a44 M44 =

(Mij denotes the i, j-th minor of the matrix A).

(A) 17 (B) 33 (C) 0 (D) -17 (E) None of the other

5. Which of the following sets is a basis of space P2 of all polynomials of degree less than
or equal to 2 ?
(A) {1 + x, 1 x, 3 + x2 } (B) {2 x, 4x x2 , 6 7x + x2 } (C) {1 + x2 , 1 x2 , 2024}
(D) {2x x2 , 1 x, x2 + 2} (E) More than one of the choices are correct

Page 1 of 9
 6. The dimension of the following subspace

U = {(w, x, y, z) : w + x + y = 0, x y + z = 0, w + 2y z = 0}

of R4 is:
(A) 4 (B) 3 (C) 2 (D) 1 (E) None of the other choices is correct

7. For problems 7-9, let R4 be the vector space over R under the following operations:

u + v = (v1 + u1 + 5, v2 + u2 7, v3 + u3 + 1, v4 + u4 2)
↵u = (↵u1 + 5(↵ 1), ↵u2 7(↵ 1), ↵u3 + (↵ 1), ↵u4 2(↵ 1))

The zero vector 0 under these operations is:

(A) (0,0,0,0) (B) (1,-1,1,-1) (C) (-5, 7, -1,2) (D) (5,-7,1,-2)
(E) None of the other choices is correct

8. For u = (0, 0, 0, 0), the additive inverse u is

(A) (0, 0, 0, 0) (B) ( 10, 14, 2, 4) (C) ( 10, 14, 2, 4) (D) (1, 1, 1, 1)
(E) None of the choices is correct

9. Let the following vectors be columns of a matrix A.

v1 = (1, 0, 1, 0), v2 = (1, 2, 0, 1), v3 = (0, 7, 1, 3), v4 = (1, 0, 1, 3)

Which of the following is a solution of homogeneous system Ax = 0.

(A) (0, 0, 0, 0) (B) (0, 2, 1, 3) (C) ( 1, 3, 0, 2) (D) (1, 1, 5, 2)
(E) None of the choices is correct

10. Let A and B be 3 ⇥ 3 matrices such that det A = 4, det B = 2, then det 2AT B 1 A3 =
(A) -1024 (B) 1024 (C) 2048 (D) -2048 (E) -256

11. Let S be the subset of all symmetric 4 ⇥ 4 matrices with real entries. This set is a subspace
of all 4 ⇥ 4 matrices with real entries. What is the dimension of this subspace?
(A) 12 (B) 16 (C) 10 (D) 6 (E) None of the other choices is correct

a b c a b c
12. If d e f =1, then find 3d 3e 3f
g h i 10g 5a 10h 5b 10i 5c
(A) 20 (B) 30 (C) 10 (D) 15 (E) None of the choices is correct

Page 2 of 9
13. Let V be a vector space consist of real valued functions defined on [0, ⇡/2], and let the norm
on V be defined as: Z ⇡/2
||f (x)|| = |f (x)|2 dx
0

p = k sin(x), find the value of k such that ||f || =p1.

If f (x)
⇡ 2 4 ⇡
(A) (B) p (C) p (D) (E) None of the choices is
4 ⇡ ⇡ 2
correct

14. Which of the following is a subspace of R3 :

(A) {(x, y, z) : x + y + z = 1} (B) {(x1 , x2 , x3 ) : x1 x2 x3 = 0} (C) {(x, y, z) :
2 2 2 2 2
y =x +z } (D) {(x, y, z) : x + y + z = 0} (E) {(x, y, 1) : y = 5x}

15. Let v i = ai î + bi ĵ + ci k̂, 1  i  3, represents the needles of a Clock at a certain time. Then
(A) v i · v j = 0 for each i 6= j. (B) v i · v j = 0 for all i, j 2 {1, 2, 3}. (C) v i ⇥ v j = 0
for all i, j 2 {1, 2, 3}. (D) v 1 ⇥ (v 2 ⇥ v 3 ) = 0. (E) None of the choices is correct

16. Let A be an n ⇥ n matrix and x1 , x2 2 Rn be distinct solutions of a homogenous system

Ax = 0. Then which of the following must always be true.
(A) Ax = b is consistent for every b 2 Rn . (B) x1 , x2 are solutions of A 1 x = 0.
(C) Rows of A are linearly independent (D) Columns of A are linearly dependent
(E) None of the choices is correct

17. Let U ⇢ R4 be the subspace of all solutions of the following system. Then a basis of U is
x + 2y + 3z + w = 0
x+z+w = 0
2x + 4y + 6z + w = 0

(A) {(1, 1, 1, 0), (1, 0, 1, 0)}.

(B) {(1, 2, 1, 1), (1, 1, 1, 3), (1, 0, 0, 1)}.
(C) {(2, 2, 2, 0), (0, 0, 0, 1), (1, 1, 1, 1)}
(D) {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}
(E) {( 1, 1, 1, 0)}

18. Let v 1 , v 2 , v 3 be vectors in R3 whose sum is zero vector. Then which of the following must
always be true.
(A) vectors are orthogonal to each other. (B) vectors are parallel to each other.
(C) vectors lie in same plane (D) vectors are linearly independent (E) None of
the choices is correct

Page 3 of 9
 2 3
1 a 1 0
19. If AT = 4 a 2 a 0 5 is an elementary matrix, then
a 1 0 a
(A) det(A) = 0 (B) RRE form of AT has a zero row (C) A is Upper Triangular
(D) A is Lower Triangular (E) A is a diagonal matrix

20. Let î, ĵ, k̂ be standard direction vectors in R3 , then the length of projection of k̂ on î is
1
(A) î (B) (C) 1 (D) 0 (E) 2
2

Page 4 of 9
Descriptive Questions

For your answers, use the space provided to you. You may only get partial or
no marks for providing just the answers without working/justification.

1. (5 points) We have already defined projection of a vector ~v on a given vector ~a as follows:

~v · ~a
proj~a (~v ) = ~a
~a · ~a
Given a linearly independent set {u~1 , u~2 , u~3 } in R3 , we can construct a new orthogonal
set {v~1 , v~2 , v~3 } as follows:
v~1 = u~1
v~2 = u~2 projv~1 (u~2 )
v~3 = u~3 projv~1 (u~3 ) projv~2 (u~3 )
[The problem starts from here]
Use the algorithm provided above to construct an orthogonal basis of R3 using the set
of vectors {(1, 2, 1), (1, 1, 3), (2, 1, 1)}.

Page 5 of 9
2. (5 points) Let v1 = (a, a + 1, a + 2), v2 = (b, b + 1, b + 2), v3 = (c, c + 1, c + 2) be vectors
in R3 , with a, b, c distinct.
(i) Is the set V = {v1 , v2 , v3 } linearly dependent?
(ii) If yes, find a non zero linear combination of v1 ,v2 ,v3 that is a zero vector.
(iii) Find a vector w such that the set {v1 , v2 , w} is a basis of R3 .

Page 6 of 9
3. (5 points) For each integer n 0, let Pn denote the vector space of all polynomials with
coefficients in R, under standard operations, and of degree less than or equal to n, i.e.,

Pn = {p(x) = a0 + a1 x + ... + an xn : a0 , ..., an 2 R}.

Show that
W = {p 2 Pn : p(0) + p(1) = 0}
is a subspace of Pn . Find a basis and dimension for W .

Page 7 of 9
4. (5 points) The figure shows a network of one-way streets with traffic flowing in directions
indicated. The flow rates along the streets are measured as the average number of vehicles
per hour. To avoid traffic jam, the inward flow at each point must be equal to outward
flow. For instance, at point A we must have x1 + x3 = 171 + 189. Find the values of
xi ’s for equal flow.

Page 8 of 9
Vector Space Axioms
For any vectors u,v,w in a real vector space V , and any real numbers k and l: 1) u + v is
in V .
2) u + v = v + u.
3) u + (v + w) = (u + v) + w.
4) There exists a zero vector 0 in V such that u + 0 = u for any vector u.
5) For any u, there exists a vector denoted by u such that u + (-u) = 0.
6) ku is in V .
7) k(u + v) = ku + kv.
8) (k + l)u = ku + lu.
9) k(lu) = (kl)u.
10) 1u = u.

Page 9 of 9