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