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Purdue University, Spring 2021Instructor: Marius Dadarlat MA265 Linear Algebra — Exam 1 Date: March 3, Spring 2021 Duration: 60 min Name: PUID: All answers must be justified and you must show all your work in order to get credit. The exam is open book. Each students should work independently, Academic integrity is strictly observed. Problem Points Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total: 100
11

MA265 Linear Algebra — Exam 1 - math.purdue.edu

Jul 31, 2022

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Page 1: MA265 Linear Algebra — Exam 1 - math.purdue.edu

Purdue University, Spring 2021 Instructor: Marius Dadarlat

MA265 Linear Algebra — Exam 1

Date: March 3, Spring 2021 Duration: 60 min

Name:

PUID:

All answers must be justified and you must show all your work in order to get credit.

The exam is open book. Each students should work independently, Academic integrityis strictly observed.

Problem Points Score

1 10

2 10

3 10

4 10

5 10

6 10

7 10

8 10

9 10

10 10

Total: 100

Page 2: MA265 Linear Algebra — Exam 1 - math.purdue.edu

1. [10pt]Find all the numbers x for which the vector

2

4�3x13

3

5 belongs to the subspace of R3

generated by the vectors

2

413�5

3

5 and

2

4�2�49

3

5.

1

I

a a

b belongs to spangui Try rank fit I 5rankfuTTT

Since rank Fir 2 as I are linearlyindependent

we must have rank vi it 53 2

rank II 1 3 t.is 11 0

o Xo naut X 15

answere 2 3

l's 7.1 1 I It t2

i'IstO O X

Page 3: MA265 Linear Algebra — Exam 1 - math.purdue.edu

2. [10pt]Consider the matrix A =

2

40 2 00 0 �10 0 0

3

5. Let S be the subspace

of R3 consisting of those vectors x such that A2x = 0. Find a basis of S.

2

i It It OL

A 1 o fo 111 1 1 HITrank 21 1

S 21 Xi xz in IRdims 1 2

basis 4 11,1 14 1x lilt

Page 4: MA265 Linear Algebra — Exam 1 - math.purdue.edu

3. [10pt]If two matrices B and C have inverses B�1 =

1 1�3 1

�and C�1 =

1 31 1

�,

and A = BC, compute A�1.

3

A e LBC c B l I 1311 1 8242

not relevant

B c l's'ell I f I I

Page 5: MA265 Linear Algebra — Exam 1 - math.purdue.edu

4. [10pt]Let L : R2 ! R3 be a linear transformation such that

L

✓10

�◆=

2

4112

3

5 , L

✓11

�◆=

2

4232

3

5 . Compute L

✓12

�◆.

4

41 2111 I 3 LILI 2117 4411

21 111 1,11 IIIlift 3 lil iii 131 111 111

LI AT The columnsof A are Lf and 491

It L

iii iii HEI III

Page 6: MA265 Linear Algebra — Exam 1 - math.purdue.edu

5. [10pt]Let A be a 4⇥5 matrix. Which of the following statements are true and which are false?Indicate clearly your answers. For this question you do not need to include explanationsfor your answers.

A. It is possible that the rank of A is 3.

B. It is possible that the null space of A has dimension 1.

C. It is possible that Ax = 0 has only one solution, the trivial solution.

D. It is possible that there exists two linearly independent vectors u and v in R5 suchthat Au = Av = 0.

E. It is possible that the columns of A are linearly independent.

5

A TRUE Ex A f I

B TRUE es Ex A ooorank 1 4

C FALSE rank Is 4 olim fulfill 754 1

then Nul I

D TRUE Ex If rank A 3 is 2 dimension

E

fl a p o

a 1 1.5 1 1

The columns of A are vectors in1124

There are at most 4 bnearlyindependent

vectors in ir4 since olim fr 41 4

Thus any 5 vectors in lR4 are linearlydependent

Page 7: MA265 Linear Algebra — Exam 1 - math.purdue.edu

6. [10pt]Let

A =

2

66664

x2 1 13 3 43 1 x 4 22 2 2 2 x3

1 3x4 4 3 112 2 4 8 1

3

77775.

The determinant of A is a polynomial in x. What is the coe�cient of x10 in this

polynomial? No use of computer software is allowed for computing this determinant.

Hint: you don’t need to compute all the terms of det(A) in order to answer the question.

6

After interchanging columns 3 pairs

outer en'tft X I I 53 2 443 I terms of lower degree

44 4

til 1411 7 Ight

til RICH 31,481t

f I Ix

24xt t

lo24 X

Answer 1247a

Page 8: MA265 Linear Algebra — Exam 1 - math.purdue.edu

7. [10pt]Let A be an invertible 5⇥ 5-matrix such that 2A2 = �AT . Compute det(A).

7

det AZ out AT

25 deter C115out I

32 detCH delCtl

A invertible outCH fo

32 outCH t 1 outCHo

32 det It I o

f det Ctl 32

Page 9: MA265 Linear Algebra — Exam 1 - math.purdue.edu

8. [10pt]Consider the vectors

u =

2

664

1000

3

775 v =

2

664

0100

3

775 w =

2

664

0010

3

775 k =

2

4001

3

5 0 =

2

4000

3

5

Let A be a 3⇥ 4 matrix such that Au = 0, Av = 0 and Aw = k. What are the possiblevalues of the rank of A?

8

rank It dim Al 4

dim ul 1 72 since it J in Nul I

are linearly independent

Thus rn t1 4 olim Nul AI E 4 2 2

rank CHEZ

since It w I o rank 71

Thus rank AI or rank 1 2

Bother

Ex Ahas rank L

A8 has rank

z

U I 1

These 2 matrices satisfythe

giver conditions

Page 10: MA265 Linear Algebra — Exam 1 - math.purdue.edu

9. [10pt]Consider a linear system whose augmented matrix can be reduced via row operations tothe echelon form 2

41 0 �2 | a0 1 a | a� 30 0 a2 � 9 | a� 3

3

5

(i) For which values of a will the system have no solution?

(ii) For which values of a will the system have a unique solution?

(iii) For which values of a will the system have infinitely many solutions?

9

a3

a I 3

a 3

l O ZUni if fgeo a

a g22 940

thus a f I 3

If a 3go 3 inhIiE

o O O 0 solutions

X 2 30 Xz S

Xz 13 3 0

1 25

Xz 35 S in 112

3 5

It a 3

o.io g i 71I euf

O Xz6

Page 11: MA265 Linear Algebra — Exam 1 - math.purdue.edu

10. [10pt]Find a matrix A with det(A) = 1 whose adjugate is adj(A) =

2

41 0 20 1 00 0 1

3

5 .

10

A outta adiI f adi 1 1

Thus A ll is the inverse of f

RREF

I it

i