MA 351: Introduction to Linear Algebra and Its ...
Transcript of MA 351: Introduction to Linear Algebra and Its ...
MA 351: Introduction to Linear Algebra and Its Applications
Fall 2021, Test One
Instructor: Yip
• This test booklet has FIVE QUESTIONS, totaling 100 points for the whole test. You have
75 minutes to do this test. Plan your time well. Read the questions carefully.
• This test is closed book, with no electronic device.
• In order to get full credits, you need to give correct and simplified answers and explain in
a comprehensible way how you arrive at them.
Name: (Major: )
Question Score
1.(20 pts)
2.(20 pts)
3.(20 pts)
4.(20 pts)
5.(20 pts)
Total (100 pts)
1
Answer key
1. Let A =
0
B@2
�1
2
3
0
�1
1
�2
5
�19
8
�15
1
7
�19
1
CA .
(a) Find an explicit, easily checkable condition on B =
0
B@a
b
c
1
CA such that AX = B is solvable.
Find also an explicit numerical example of B such that AX = B is not solvable.
(b) Find a basis for Col(A) by selecting appropriate columns of A. Write also the columns
of A that are not used as as a linear combination of the basis vectors.
(c) Find a basis for Null(A).
2
see also Spring2019 Midterm 1 1
123I 19 i a
t o 2 s t b 128 9587 by2 I 5 15 19 c
2 I 5 15 19 C
10 3 3 315 atsbTake
e.gI it c 10 2 8
71b so that
j
O 1 1 1 5 cab
o 2 8
71b
I I I 5 9 263O O O O O Cab 1 a
So we need Ct abt 931 0 ie at fb 130 0
You can use this blank page.
3
b Consider dependency relation for thecols ofA
Ci Nit G Nat GUs 4214 8 45 0
go l
O O O O O
pivots FHence throw away
Us V4 Us of A and keep a in
Col1A Span E LIbasisofcollA
C3 4 Cy f E y Cr 22 8pityCz L t p 58so that
2 8 78 u Lt P 58742 243 44 840
L I p 0,2 0 M3 2M Ma
Ho Ba F 044 Su Uz
d O BO 8 1 V5 74 542
You can use this blank page.
4
9 NullIA X AX O
8pivots
43 4 XEB Xe J X 22 88 72X2 L t p 58
I FI rY
1 11 44s s
Basis
2. Consider the following 3⇥ 3 system of linear equation:
0
B@1
3
4
2
�1
1
�3
5
(a2 � 14)
1
CA
0
B@x
y
z
1
CA =
0
B@4
2
a+ 2
1
CA .
where a is some parameter.
Find the value(s) of a such that the system has (i) unique solution, (ii) infinitely many
solutions, and (iii) no solution.
5
seealso Spring2016Midterm 1 3,4 Hw3 AddProb 2
Ii la lo ly3 I 5 20 7 a 2 a 14
oo o a b a 4
is Unique solution a 16 0 at 4 4
ii inf many solus a b of a 5a 4
iii Nosoln a I ta 440
AI
3. Consider the following 2⇥ 2 system of linear equation:
x+ (�� 3)y = 0,
(�� 3)x+ y = 0,
where � is a parameter. For what values of � is such that the system has infintely many
solutions?
7
lis il p ly
II
Forinfinitely manysolutions we need
I A35 0 I 4 3 11
4 401
4. You are given the list of vectors {X1, X2, X3} which is linearly independent. For each of the
following lists, investigate if it is linearly (in)dependent. If dependent, write down an example
of linear relationship between the vectors.
(a) {Y1, Y2} where Y1 = X1 +X2, Y2 = X1 �X2.
(b) {Z1, Z2, Z3} where Z1 = X1 +X2 +X3, Z2 = X1 �X2 + 2X3, Z3 = 2X1 �X2 �X3.
(c) {W1,W2,W3} where W1 = X1 �X2 +X3, W2 = X1 +X2 + 2X3, W3 = X1 � 3X2.
9
Penney p 111 2.11 2.13 2.17 2.18
a cilitczyz o e Ci xx tea Xi xp o
let c Xi tf a XeoSince X X him ind Cites o 9 0
C 9 0 2 0
Hence
Y.Yzaelinind.chchitchat 323 0 9 0 9 0 Go
d XXX E Xi Xt 243 5 24 42 137 0
EpaX 1 34 2 9 D
1
You can use this blank page.
10
o 3 90 1 3 0 0 2 3 0
fo 3 f notreevar0 0 9 0 Hence 2 22,23 linind
C C W tEWstGWz 0
C Xi XztX3 te xxx 243 15 4 3123 0
eggslit kg3flXzt 5EX3oCSimeXyXz.Xlining
fi 1 it0 I 1 O
8 o i g www.wO 0 0
Tree find
You can use this blank page.
11
Cz d Ca d C 2x
4 Wit LW 2W O
2WitWaWtCheck
Xi Xt Xz XtXzt2X 41 3 12 0
5. Suppose the solution of a 2⇥ 4 system AX = B is given by
X =
0
BBBB@
0
1
1
0
1
CCCCA+ s
0
BBBB@
1
1
2
0
1
CCCCA+ t
0
BBBB@
1
0
�1
3
1
CCCCA
where s and t are free variables.
Find A and B. Is there a unique answer? If not, write down two explicit examples.
12
See Homework 4 p88 106 107
r espanning
vectors for Wall sp of A
A a b c de f g n
f a Esco
If 8 acne
1 y0 1 3 3 O
C d d pa L 3Bo i 3 ob 3 a 38Fu
You can use this blank page.
13
La fro fa b e d 1 3 1 o
d o
pla b c d 3 3 0 1
Have A 33At B 33897 1 11
Make use of the translation victory3 1 p p 2
3 08 8 3
Hence an example of AX B is
t FE
You can use this blank page.
14
Another example multiply first row by 2
C DEDEor add the 2 rows together
I Y E Ez o