BRAC University Course Code: MAT 216 Home work* Sheet # 1 following matrix equation equation for a, b, c and d . 1. Solve the following b+c ⎤ ⎡ a−b ⎡8 1 ⎤ ⎢3d + c 2a − 4d ⎥ = ⎢7 6⎥ . ⎣ ⎦ ⎣ ⎦
2. Consider the matrices : 0⎤ ⎡3 ⎢ ⎥ A = −1 2 ⎢ ⎥ ⎢⎣1 1 ⎥⎦
⎡4 − 1⎤ ⎥ , C= 0 2 ⎣ ⎦
, B =⎢
⎡ 1 5 2⎤ ⎢ ⎥ D = −1 0 1 , ⎢ ⎥ ⎢⎣ 3 2 4⎥⎦
⎡1 4 2⎤ ⎢3 1 5 ⎥ , ⎣ ⎦
⎡ 6 1 3⎤ ⎢ ⎥ E = − 1 1 2 , ⎢ ⎥ ⎢⎣ 4 1 3⎥⎦
Compute the following (where possible) (a) D (a) D + E (b) –7C b) –7C , (c) 2B (c) 2B –C, (d) –3 (d) –3 (D + 2E) , (e) A-A (e) A-A , (f) tr (f) tr (D – 3E).
3. Using the matrices in exercise (2) , (2) , compute the following (where possible) : T
T
T T
T
T
T
(a) 2A (a) 2A + C, (b) (b) (2E – 3D ) , (c)( (c)( D – E ) , (d) B (d) B + 5C , (e)
1 2
T
C -
1 4
A.
4. Using the matrices in exercise (2) , compute the following (where possible) . T (a) AB , (b) BA , (c) (3E) D , (d) (AB )C , (e) A (BC) , (f) (DA) , T
T
T
T
(g) (C B) A , (h) tr (DD ) , (i) tr (4E – D).
5. Using the matrices in exercise (2) , compute the following (where possible) : T
T
T
(a) (2D – E ) A , (b) ( B A – 2C ) .
*These problems are for the students only as home work. Search the reference books for more examples.
BRAC University Course Code: MAT 216 Home work* Sheet # 2 1.
Solve each of the following systems by Gaussain elimination or Gauss - Jordan elimination: x1 + x2 + 2 x3 = 8
x − y + 2 z − w = −1
2 x1 + 2 x2 + 2 x3 = 0
(i) − x1 − 2 x2 + 3 x3 = 1 (ii) 3 x1 − 7 x2 + 4 x3 = 10
− 2 x1 + 5 x2 + 2 x3 = 1 8 x1 + x2 + 4 x3 = −1
(iii)
2 x + y − 2 z − 2 w = −2
− x + 2 y − 4 z + w = 1 − 3w = −3 3 x
2. Solve each of the following homogeneous system of linear equations by Gaussain elimination or Gauss - Jordan elimination: x5 = 0 2 x1 + 2 x2 − x3 + 2 x + 2 y + 4 z = 0
(i)
− x1 − x2 + 2 x3 − 3 x4 + x5 = 0 x1 + x2 − 2 x3
− x5 = 0
(ii)
x3 + x4 + x5 = 0
− y − 3 z = 0 2w + 3 x + y + z = 0 − 2 w + x + 3 y − 2 z = 0 w
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: x + y − z = 1 2 x + 3 y + λ z = 3
.
x + λ y + 3 z = 2
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 : x + y + z = 6 x + 2 y + 3 z = 10 . x + 2 y + λ z = μ
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) λ x + y + z = 1 x + λ y + z = 1 x + y + λ z = 1
− 3 z = −3 (iii) 2 x + λ y − z = −2 x + 2 y + λ z = 1
(ii )
x + y + kz = 2 3x + 4y + 2z = k 2x + 3y –z = 1 x + y + λ z = 1
x
(iv)
x + λ y + z = λ 2 λ x + y + z = λ
*These problems are for the students only as home work. Search the reference books for more examples.
BRAC University Course Code: MAT 216 ⎡ 1 −2 3 ⎤ ⎢ ⎥ 6. Let A = 6 7 −1 , ⎢ ⎥ ⎢⎣− 3 1 4 ⎥⎦ (a) Find all the minors of A (b) Find all the cofactors of A, (c) Find adj (A) , (d) Find A
7.
-1
-1
, Using A =
1 det( A)
adj (A).
Find the inverse of the following matrices if it exists, using [ A: I]:
⎡ 2 5 5⎤ ⎢ ⎥ (i) − 1 − 1 0 ⎢ ⎥ ⎢⎣ 2 4 3⎥⎦ ⎡1 ⎢2 (v) ⎢ ⎢1 ⎢ ⎣1
3 1
⎡2 − 3 5 ⎤ ⎢ ⎥ (ii) 0 1 − 3 ⎢ ⎥ ⎢⎣0 0 2 ⎥⎦
⎡ − 1 2 − 3⎤ ⎡3 4 − 1⎤ ⎢ ⎥ ⎢ ⎥ (iii) 2 1 0 (iv) 1 0 3 ⎢ ⎥ ⎢ ⎥ ⎢⎣ 4 − 2 5 ⎥⎦ ⎢⎣2 5 − 4⎥⎦
1⎤
⎡1 − 1 1⎤ ⎡− 1 2 − 3⎤ ⎥ 5 2 2 ⎥ ⎥ (vi) ⎢2 − 1 0⎥ (vii) ⎢ 2 1 0 (viii) ⎢ ⎥ ⎢ ⎥ 3 8 9⎥ ⎢ ⎥ ⎢ ⎥ ⎣1 0 0⎦ ⎣ 4 − 2 5 ⎥⎦ 3 2 2⎦
⎡1 1 1⎤ ⎢ ⎥ 8. If A = 1 2 3 & B = ⎢ ⎥ ⎢⎣1 4 9⎥⎦
⎡1 − 1 2 ⎢3 0 2 ⎢ ⎢2 1 − 1 ⎢ ⎣1 0 1
1⎤
⎥ ⎥ 1⎥ ⎥ 1⎦ 2
⎡ 2 5 3⎤ ⎢3 1 2⎥ , prove that ( AB )−1 = B −1. A−1 ⎢ ⎥ ⎢⎣1 2 1⎥⎦
9. Solve the following system of linear equations using x = A-1 b
(i)
x1 + 3 x2 + x3 = 4
5 x1 + 3 x2 + 2 x3 = 4
2 x1 + 2 x2 + x3 = −1
3 x1 + 3 x2 + 2 x3 = 2
2 x1 + 3 x2 + x3 = 3
(ii)
x2 + x3 = 5
x + y + z = 5
(iii) x + y − 4 z = 10
− 4 x + y + z = 0
*These problems are for the students only as home work. Search the reference books for more examples.
BRAC University Course Code: MAT 216 Home work* Sheet # 3 1. Verify whether the following sets are subspace of R 3 / R 4 or not. (i) S = {(x, 2y, 5) : x,y e R } (ii) S = {(x, x + y, 3z) : x, y, z e R } (iii) S = {(x, y, z) e R 3 : x - y + z = 0 } 4 (iv) S = {(x, y, z, t) e R : 3x - 2y - 2z - t = 0 }
(v) S = {(x, y, z) e R 3 : 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). 3 4. Determine whether or not the following set of vectors span 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)} .
3
4
6. Determine whether each of the following sets form a basis for R / 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)} . *These problems are for the students only as home work. Search the reference books for more examples.
BRAC University Course Code: MAT 216
7. Find a basis for the row space, a basis for the column space and the rank of the following matrices: ⎡2 − 1 3 4⎤ ⎡6 2 0 4 ⎤ ⎡ 1 2 0 − 1⎤ ⎢0 3 4 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ . (i) A = − 2 − 1 3 4 (iii) A = ⎢ ⎢ ⎥ (ii) A = ⎢ 3 4 1 2 ⎥ ⎢2 3 7 5 ⎥ ⎢⎣ − 1 − 1 6 10 ⎥⎦ ⎢⎣− 2 3 2 5 ⎥⎦ ⎢ ⎥ ⎣2 5 11 6⎦
⎡0 ⎢1 (iv) A = ⎢ ⎢3 ⎢ ⎣1
− 3 − 1⎤ 0 1 1⎥ ⎥ 1 0 2⎥ ⎥ 1 −2 0 ⎦ 1
⎡1 ⎢1 (v) A = ⎢ ⎢2 ⎢ ⎣3
−2 4 3 −1 3 −4 −7 8 1 −7 3
1
− 3⎤ − 4⎥⎥ . − 3⎥ ⎥ − 8⎦
8. Find a basis for the Null space, the rank and the Nullity of the following matrices: ⎡−1 2 0 4 5 − 3⎤ ⎡1 −1 3 ⎤ ⎡ 1 4 5 2⎤ ⎢ 3 −7 2 0 1 ⎥ 4 ⎥ (ii) A = ⎢5 − 4 − 4⎥ (iii) A = ⎢ 2 1 3 0⎥ (i) A = ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ 2 −5 2 4 6 1⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎣7 − 6 2 ⎦ ⎣− 1 3 2 2⎥⎦ − − − 4 9 2 4 4 7 ⎣ ⎦ ⎡ 1 ⎢ 0 ⎢ (iv) A = ⎢ 2 ⎢ ⎢ 3 ⎢⎣ − 2
−3
2
2
3
6
0
1 ⎤
− 3 ⎥⎥ 4 ⎥ −3 −2 4 ⎥ −6 0 6 5 ⎥ 9 2 − 4 − 5 ⎥⎦
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)}.
3
10. Let U be the subspace of 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 .
4 11. Let W be the subspace of 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.
*These problems are for the students only as home work. Search the reference books for more examples.
BRAC University Course Code: MAT 216 Home work* Sheet # 4 1. Determine whether each of the following Transformation is a linear t ransformation: (i) T ( x, y, z ) = ( x − y, x − z ) (ii) T ( x, y, z ) = (3 x − 2 y + z, x − 3 y − 2 z ) (iii) T ( x, y, z ) = ( x + 1, y + z ) . (iv) T ( x, y ) = ( x + y, 1) 2. Let T : R → R be the linear transformation defined by T ( x, y, z, t ) = ( x − y + z + t , x + 2 z − t , x + y + 3 z − 3t ) . Find a basis & dimension of the range space of (T) & the null space of (T). 4
3
3 3 3. Let T : R → R be the linear transformation defined by T ( x, y, z ) = ( x + 2 y − z , y + z, x + y − 2 z ) . Find a basis & dimension of (i) Range(T) & (ii) Ker (T).
4. Let T : R → R be the linear transformation defined by T ( x, y, z ) = (3 x − y, y − z,3 x − 2 y + z ) , Find a basis & dimension of (i) Range(T) & (ii) Ker (T). 3
3
5. Let T : R → R be the linear transformation defined by T ( x, y, z ) = ( x + 2 y − 3 z,2 x − y + 4 z ,4 x + 3 y − 2 z ) , Find a basis & dimension of (i) Range (T) & (ii) Ker (T). 3
3
6. Find all eigenvalues and the corresponding eigenvectors of the following matrices:
⎡1 2 − 1⎤ ⎢ ⎥ (i) A = 0 − 2 0 ⎢ ⎥ ⎢⎣0 − 5 2 ⎥⎦
⎡2 − 2 1 ⎤ ⎢ ⎥ (ii) A = 2 − 8 − 2 ⎢ ⎥ ⎢⎣1 2 2 ⎥⎦
⎡ 3 2 4⎤ ⎢ ⎥ (iii) A = 2 0 2 ⎢ ⎥ ⎢⎣4 2 3⎥⎦
7. Find a matrix P that diagonalizes the following matrices , also find ⎡ − 1 − 2 − 2⎤ ⎡1 ⎡ − 14 12⎤ ⎥ ⎢ ⎢ (iii) A = 2 2 1 (i) A = ⎢ (ii) A = 1 ⎥ ⎢ ⎥ ⎢ ⎣− 20 17 ⎦ ⎢⎣− 1 − 1 0 ⎥⎦ ⎢⎣− 1
P − AP : 4 1⎤ 1
⎥ ⎥ 3 1⎥⎦
1
0
Solve the following p roblems giv en in the book " Elementary linear algebra by Howard 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 their corresponding Eigenvectors that form bases for eigenspace. If possible, diagonalize those matrices. *These problems are for the students only as home work. Search the reference books for more examples.
BRAC University Course: MAT-216(Mathematics III) Homework Sheet (Calculus) # 4
1. Evaluated the iterated integrals: π
(a)
x
2
1
∫∫ π
cos
x
0
1 1
y
dy dx (b)
x
∫ ∫ ( xy
2 y
x
0 0
+
dy dx (c) 2
1)
x
2
∫∫
e
4− x 2
2
y 2
dx dy (d)
∫ ∫ 0
1 0
e
x 2 + y 2
dy dx .
0
2
2. (a) Find the area of the region inside the circle r = 4sin θ and outside the circle r = 2. (b)
∫∫ x
1 2
y 2
1 y=0,
+
R
dA , where R is the sector in the first quadrant bounded by
+
2
y=x
2
and x + y = 4 . a
3. Use polar coordinates to evaluate the double integral
a 2 − x 2
∫ ∫ ( x −
a
2
+
y
) dydx .
2 12
0
2
4. (a) Find the volume of the solid that is bounded by the cylinder y = x and by the planes y + z = 4
and
z =0.
(b) Find the volume of the surface enclosed by the surfaces � � � and � � − − .
5. Evaluated the iterated integral by converting to polar coordinates: 1
(a)
1 − x 2
∫ ∫ ( x 0 a
(c)
+
2
y )dydx 2
a
−
2 x − x 2
∫ ∫
(b)
0 2
∫ ∫ 0
2
0 x
0
2
dydx
(1
+
x
2
4
y = 0
where u (b) Evaluate
=
2
x =
2 x − y
dxdy by applying transformation T :
2
y
0 , x − y
2
y
2 +
0
1
2 x − y
∫∫ x
dydx
+
x − y
R
=
y
y 2
)
y
x =
+
(a > 0) .
3 2 2
∫ ∫
6. (a) Evaluate
x − y
+
x 2
,
v
=
y
2
and integrating over an appropriate region in uv–plane.
dA , where R is the region enclosed by the lines
=
1 , x + y
=
1 & x + y
=
3 , using the transformation .
BRAC University Home Work sheet # 5 MAT – 216
∫ ( xy + z
1. Evaluate the line integral
3
) ds from (1,0,0) to (-1,0,0) along the helix C that is
C
represented by the parametric equation x = cost, y = sin t , z = t (0 ≤ t ≤ π ) .
∫
2. Evaluate xy dx + x 2 dy if C
(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). 2
(c) Parametric equation for C are x = 3t – 1, y = 3t – 2t ; 1 ≤ t ≤ 5 3 . 3. Show that (a)
∫ (6 x y − 3 xy 2
the points (1,2) and (3,4) 4. Let
2
) dy + (6 xy 2 − y 3 ) dx is independent of the path joining
(b) hence evaluate the integral.
F ( x, y ) = (3 x 2 y + 2) i + ( x 3 + 4 y 3 ) j
Determine if
∫ F . dr
represents a force field.
is independent of path if it is, find a potential function φ .
C
5. Let F ( x, y ) = 2 x y 3 i + (1 + 3 x 2 y 2 ) j (a) Show that F is a Conservative Vector field on the entire xy – plane , (b) find f by first integrating (c) find f by first integrating
∂ f ∂ x ∂ f ∂ y
, .
6. Use the potential function obtained in example (5) to evaluate the integral (3,1)
∫(
1, 4 )
3
2 2 2 xy dx + (1 + 3 x y ) dy .
th
From Book :- (Calculus, Howard Anton 10 edition, soft copy)
BRAC University Homework sheet # 6 MAT – 216
Fourier Series and application 1.(a) Determine the Fourier series for
− x ,
f ( x ) =
−
x ,
(b)Find the Fourier coefficients for
0 , − 5 < x < 0 f ( x ) = 3 , 0 < x < 5
4 ≤ x ≤ 0 , 0 ≤ x ≤ 4
Period = 8
Period = 10 .
f ( x ) = x , 0 < x
2. Expand
(i) Sine series
<
2 in a half – range
(ii) Cosine series.
1 1 4 − x , 0 < x < 2 3. Expand f ( x ) = , in a Fourier series of Sine terms only. 3 1 x − , < x < 1 4 2 4. Graph each of the following functions and find its corresponding Fourier series, using properties of even and odd function wherever applicable.
8 , 0 < x < 2 (a) f ( x ) = , Period 4 x 8 , 2 4 − < <
− x,
(b) f ( x ) =
− 4 ≤ x ≤
x, 0 ≤ x ≤ 4
5. Expand f ( x ) = cos x
0
, Period 8
(c) f ( x ) = 4 x
< 10 , Period 10 ,0 < x
2 x, 0 ≤ x ≤ 3
(d) f ( x ) =
x, − 3 ≤ x ≤ 4
,0 < x < π in a Fourier sine series.
, 0 < x < 4 x 6. Expand f ( x ) = 8 − x , 4 < x < 8
in (a) Sine series (b) Cosine series.
, Period 6
Exercise set 15.3 - (1-6), (9-14)
Green’s theorem Exercise set 15.4 - 1-14
Double Integral
Exercise- 14.1-
1-16.
Exercise- 14.2-
1-26.
Exercise- 14.3-
1-12, 23-34.
Surface Area from Double Integral Exercise- 14.4-
1-9.
Triple Integral Exercise- 14.5-
1-12, 15-18.
Change of variables Exercise- 14.7-
1-12, 21-24, 35-37.
th
Book: Elementary Calculus- Howard Anton (10 Edition), Soft Copy