HWsheet 1-4 Linear Algebra

8/19/2019 HWsheet 1-4 Linear Algebra

1/16

BRAC University

Course Code: MAT 216

*These problems are for the students only as home work. Search the reference books for

more examples.

Home work* Sheet # 1

1. Solve the following matrix equation for a, b, c and d .

⎥⎦

⎤

⎢⎣

⎡

−+

+−

d acd

cbba

423 = ⎥⎦

⎤

⎢⎣

⎡

67

18

.

2. Consider the matrices :

A =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

1

2

0

1

1

3

, B = ⎥⎦

⎤⎢⎣

⎡ −

20

14 , C = ⎥

⎦

⎤⎢⎣

⎡

5

2

1

4

3

1 ,

D =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

423

101

251

, E =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

314

211

316

,

Compute the following (where possible)

(a) D + E (b) –7C , (c) 2B –C, (d) –3 (D + 2E) , (e) A-A , (f) tr (D – 3E).

3. Using the matrices in exercise (2) , compute the following (where possible) :

(a) 2AT + C, (b) (2E

T – 3D

T)

T, (c)( D – E )

T , (d) B

T + 5C

T , (e)

2

1C

T-

4

1A.

4. Using the matrices in exercise (2) , compute the following (where possible) .

(a) AB , (b) BA , (c) (3E) D , (d) (AB )C , (e) A (BC) , (f) (DA)T ,

(g) (CT B) A

T , (h) tr (DD

T) , (i) tr (4E

T – D).

5. Using the matrices in exercise (2) , compute the following (where possible) :

(a) (2DT – E ) A , (b) ( B A

T – 2C )

T .


2/16

BRAC University



more examples.


1. Solve each of the following systems by Gaussain elimination or Gauss - Jordanelimination:

(i)

10473

132

82

321

321

321

=+−

=+−−=++

x x x

x x x

x x x

(ii)

148

1252

0222

321

321

321

−=++

=++−=++

x x x

x x x

x x x

(iii)

333

142

222212

−=−

=+−+−

−=−−+

−=−+−

w x

w z y x

w z y xw z y x

2. Solve each of the following homogeneous system of linear equations by Gaussainelimination or Gauss - Jordan elimination:

(i)

002

032

022

543

5321

54321

5321

=++

=−−+

=+−+−−

=+−+

x x x x x x x

x x x x x

x x x x

(ii)

0232032

03

0422

=−++−

=+++

=−−

=++

z y xw z y xw

z yw

z y x

3. Determine the values of parameter λ , such that the following system has

(i) no solution (ii) a unique solution (iii) more than one solution:

23

332

1

=++

=++

=−+

z y x

z y x

z y x

λ

λ .

4. Determine the values of parameters λ & μ , such that the following system has

(i) no solution (ii) a unique solution (iii) more than one solution :

μ λ =++

=++

=++

z y x

z y x

z y x

2

1032

6

.

5. Determine the values of parameter (s) such that the following system has

(i) no solution (ii) a unique solution (iii) more than one solution :

(i) 1=++ z y xλ (ii ) x + y + kz = 2

1=++ z y x λ 3x + 4y + 2z = k

1=++ z y x λ 2x + 3y –z = 1

(iii)

12

2233

=++

−=−+−=−

z y x

z y x z x

λ

λ (iv)

2

1

λ λ

λ λ λ

=++

=++=++

z y x

z y x z y x


3/16

BRAC University



more examples.

6. Let A =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

−

413

176

321

,

(a) Find all the minors of A

(b) Find all the cofactors of A,(c) Find adj (A) ,

(d) Find A-1

, Using A-1

=)det(

1

A adj (A).

7. Find the inverse of the following matrices if it exists, using [A: I]:

(i)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

342

011

552

(ii)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

200

310

532

(iii)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−

524

012

321

(iv)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

452

301

143

(v)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

2231

9831

2252

1131

(vi)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

001

012

111

(vii)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−

524

012

321

(viii)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−

1101

1112

2203

1211

8. If

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

941

321

111

A &

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

121

213

352

B , prove that ( ) 111 . −−− = A B AB

9. Solve the following system of linear equations using x = A-1

b

(i)332

122

43

321

321

321

=++

−=++

=++

x x x

x x x

x x x

(ii)5

2233

4235

32

321

321

=+

=++

=++

x x

x x x

x x x

(iii)

04

104

5

=++−

=−+

=++

z y x

z y x

z y x


4/16

BRAC University



more examples.


1. Verify whether the following sets are subspace of 3 R / 4 R or not.

(i) S = {(x, 2y, 5) : x,y e R }

(ii) S = {(x, x + y, 3z) : x, y, ze

R }(iii) S = {(x, y, z) e

3 R : x - y + z = 0 }

(iv) S = {(x, y, z, t) e 4 R : 3x - 2y - 2z - t = 0 }

(v) S = {(x, y, z) e 3 R : x + y + z = 0 }

2. Write the vectors (1, 0, 0) and (0, 0, 1) as linear combinations of vectors

{(1, 0, -1), (0, 1, 0), (1, 0, 1)}

3. Determine whether or not,

(i) the vector (1, 2, 6) is a linear combination of (2, 1, 0), (1, -1, 2) & (0, 3, -4). (ii) the vector (1, 1, 1) is a linear combination of (2, 1, 0), (1, -1, 2) & (0, 3, -4).

(iii) the vector (3, 9, -4, -2) is a linear combination of (1,-2, 0, 3), (2, 3, 0, -1) & (2, -1, 2, 1).

(iv) the vector (2, 3, -7, 3) is a linear combination of (2, 1, 0, 3), (3, -1, 5, 2) & (-1, 0, 2, 1).

4. Determine whether or not the following set of vectors span3 R ,

(i) {(1, 1, 2), (1, -1, 2), (1, 0, 1)}

(ii) {(-1, 1, 0), (-1, 0, 1), (1, 1, 1)}

(iii) {(2, 1, 2), (0, 1, -1), (4, 3, 3)}

5. Determine whether each of the following sets are linearly independent or dependent:

(i) {(2 ,1 ,2) , (0 , 1 , - 1) , (4 , 3 , 3)} .

(ii) {(3 , 0 , 1 , -1) , (2 , -1, 0 , 1) , (1 , 1 , 1 , -2)} .

(iii) {(1 , - 4 , 2) , (3 , - 5 , 1) , (2 , 7 , 8) , (- 1 ,1 , 1)} .

(iv) {(0 , 1 , 0 , 1) , (1 , 2 , 3 , -1) , (8 , 4 , 3 , 2) , (0 , 3 , 2 , 0)} .

(v) {(1 , 3 , 2) , (1 , -7 , - 8) , (2 , 1 , - 1)} .

(vi) {(3 , 0 , 4 , 1) , (6 , 2 , -1 , 2) , (-1 , 3 , 5 , 1) , (- 3 , 7 , 8 , 3)}

(vii) {(4 , -4 , 8 , 0) , (2 , 2 , 4 , 0) , (6 , 0 , 0 , 2) , (6 , 3 , -3 , 0)} .

6. Determine whether each of the following sets form a basis for3

R /4

R :

(i) {(1 , 2 , 0) , (0 , 5 , 7) & (-1 , 1 , 3)}.

(ii) {(2 , 0 , 1) , (1 , 1 , 1)} .

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


5/16

BRAC University



more examples.

7. Find a basis for the row space, a basis for the column space and the rank of the following

matrices:

(i)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

−

−=

10

44

6

30

1

12

1

26

A (ii)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

−

=

5

21

2

10

3

42

2

31

A (iii)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

=

61152

5732

1430

4312

A .

(iv)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−−

=

0211

2013

1101

1310

A (v)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−

−

−

−

−

−

−

−=

8

3

4

3

7

7

1

2

1

4

3

1

8

3

4

3

3

2

1

1

A .

8. Find a basis for the Null space, the rank and the Nullity of the following matrices:

(i)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

−−−

−

−−

=

7

1

43

4

6

15

4

4

04

2

2

20

9

5

72

4

2

31

A (ii)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−

−

=

267

445

311

A (iii)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

=

2

0

2

2

3

5

3

1

4

1

2

1

A

(iv)

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−

−

−

−

−

=

54292

5

4

3

1

6

4

0

2

0

2

6

2

6

3

3

3

3

2

0

1

A

9. Find a basis and dimension of the subspace generated by the set of vectors

S = {(1 , 2 , 1) , (3 , 1 , 2) , (1 , -3 , 4)}.

10. Let U be the subspace of3

R spanned (generated) by the set of vectors

S = {(1 , 2 , 1) , (0 , - 1 , 0) & (2 , 0 , 2)}. Find a basis and dimension of U .

11. Let W be the subspace of4 R generated by the set of vectors

S = {(1 ,- 2 , 5 ,-3) , (2 , 3 , 1 , - 4) & (3 , 8 , - 3 , - 5)}. Find a basis and dimension of W.


6/16

BRAC University



more examples.


1. Determine whether each of the following Transformation is a linear transformation:

(i) ( ) ( ) z x y x z y xT −−= ,,, (ii) ( ) ( ) z y x z y x z y xT 23,23,, −−+−= (iii) ( ) ( ) z y x z y xT ++= ,1,, . (iv) ( ) ( )1,, y x y xT +=

2. Let T :34

R R → be the linear transformation defined by( ) ( )t z y xt z xt z y xt z y xT 33,2,,,, −++−+++−= .

Find a basis & dimension of the range space of (T) & the null space of (T).

3. Let T :33 R R → be the linear transformation defined by

( ) ( ) z y x z y z y x z y xT 2,,2,, −++−+= .Find a basis & dimension of (i) Range(T) & (ii) Ker (T).


( ) ( ) z y x z y y x z y xT +−−−= 23,,3,, ,Find a basis & dimension of (i) Range(T) & (ii) Ker (T).


( ) ( ) z y x z y x z y x z y xT 234,42,32,, −++−−+= ,Find a basis & dimension of (i) Range (T) & (ii) Ker (T).

6. Find all eigenvalues and the corresponding eigenvectors of the following matrices:

(i)

⎥

⎥⎥

⎦

⎤

⎢

⎢⎢

⎣

⎡

−

−

−

=

250

020

121

A (ii)

⎥

⎥⎥

⎦

⎤

⎢

⎢⎢

⎣

⎡

−−

−

=

221

282

122

A (iii)

⎥

⎥⎥

⎦

⎤

⎢

⎢⎢

⎣

⎡

=

324

202

423

A

7. Find a matrix P that diagonalizes the following matrices , also find APP1− :

(i) ⎥⎦

⎤⎢⎣

⎡

−

−=

1720

1214 A (ii)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−−

=

011

121

221

A (iii)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

=

131

012

141

A

Solve the following problems given in the book " Elementary linear algebra byHoward Anton and Chris Rorres, Application version, eigth edition."

_ Ex 5.5: 3(a,b), 6(a,b), 7(a,b),8(a, b,c), 11(a,c),12(a,b) _ Ex 5.6: 1, 2(a,b,c) _ Ex 8.1: 13, 16 _ Ex 8.2: 3,4, 10, 11 _ Ex 7.1: Consider the matrix given in 4(a,c,d). Find the eigenvalues and theircorrespondingEigenvectors that form bases for eigenspace. If possible, diagonalize those matrices.


7/16


8/16

BRAC University

Home Work sheet # 5

MAT – 216

1. Evaluate the line integral ds z xy

C

∫ + )(3

from (1,0,0) to (-1,0,0) along the helix C that is

represented by the parametric equation x = cost, y = sin t , z = t ( )π ≤≤ t 0 .

2. Evaluate dy xdx xyC

2+∫ if

(a) C consists of line segments from (2,1) to (4,1) and from (4,1) to (4,5).

(b) C is the line segment from (2,1) and (4,5).

(c) Parametric equation for C are x = 3t – 1, y = 3t2 – 2t ; 351 ≤≤ t .

3. Show that (a) ∫ −+− dx y xydy xy y x )6()36(3222

is independent of the path joining

the points (1,2) and (3,4) (b) hence evaluate the integral.

4. Let ( ) ( ) ( ) j y xi y x y xF 332 423, +++= represents a force field.

Determine if ∫C

dr F . is independent of path if it is, find a potential function φ .

5. Let ( ) ( ) j y xi y x y xF 223 312, ++=

(a) Show that F is a Conservative Vector field on the entire xy – plane ,

(b) find f by first integrating x

f

∂

∂

,

(c) find f by first integrating y

f

∂

∂ .

6. Use the potential function obtained in example (5) to evaluate the integral

( )

( )( ) dy y xdx xy 22

31,3

4,1312 ++∫ .

From Book :- (Calculus, Howard Anton 10th

edition, soft copy)


9/16

BRAC University

Homework sheet # 6

MAT – 216

Fourier Series and application

1.(a) Determine the Fourier series for

8 = Period

,40,

04,

)(

≤≤

≤≤−−

= x x

x x

x f

(b)Find the Fourier coefficients for

( )

. 10= Period

50,3

05,0


10/16

Exercise set 15.3 - (1-6), (9-14)


11/16

Green’s theorem

Exercise set 15.4 - 1-14


12/16

Double Integral

Exercise- 14.1- 1-16.


13/16

Exercise- 14.2- 1-26.


14/16

Exercise- 14.3- 1-12, 23-34.

Surface Area from Double Integral

Exercise- 14.4- 1-9.


15/16

Triple Integral

Exercise- 14.5- 1-12, 15-18.

Change of variables

Exercise- 14.7- 1-12, 21-24, 35-37.


16/16

Book: Elementary Calculus- Howard Anton (10

th Edition), Soft Copy

HWsheet 1-4 Linear Algebra

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