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