Exercises Môn Applied Linear Algebra | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

lOMoAR cPSD| 59085392
Exercises on Applied Linear Algebra Duong Thanh PHAM August 29, 2021
1 Systems of Linear Equations and Matrices
1.1 Systems of linear equations and Gaussian elimination
Exercise 1. Solve the following systems
2x1 − x2 − x3 = 4 x2x 1. 3x 11+ 2− xx22 + 3−
1 + 4x2 − 2x3 = 11
3x1 − 2x2 + 4x3 = 11
2xx33 −− 23xx44 = 6= 4 3.
x1 + x2 + 2x3 = −1
32xx11 + 2− 3xx22 −+ 2xx33+ 2+ xx44 = 4= −8
2. 2x1 − x2 + 2x3 = −4
4x1 + x2 + 4x3 = −2
Exercise 2. Solve the following systems
mx1 + x2 + x3 = 1
(m + 3)x1 + x2 + 2x3 = m 2. mx 1. x
1 + (m − 1)x2 + x3 = 2m
1 + mx2 + x3 = m
x1 + x2 + mx3 = m2
3(m + 1)x1 + mx2 + (m + 3)x3 = 3
Exercise 3. Solve the following systems by using Gaussian Elimination
3xx++ 2y +y z++z t++t u−= 2xx+ 2− 73u = −2 1.
2yy−+zz+−tt−+2uu = 1= 1 3.
y5x+ 2+ 4zy+ 2+ 3t z+ 6+ 3ut= 23− u = 12
42xx −− 1014yy +
5+ 7zz −− 57tt + 7+ 11uu= 1= −1
x2x−+yy+−zz+−tt−+2uu = 1= 0 2. lOMoAR cPSD| 59085392
43xx + 5+ 3yy −− 53zz −− 53tt + 7+ 4uu = 3= 2 32xx +− yy
−+ 72zz +− t3t−+ 5u = 1u = 2 4.
x3x+ 3− y2y−+
72zz+ 5− t5t−+ 87uu= 3= 3
Exercise 4. Solve the corresponding homogeneous systems of the systems in Exercise ??. 1.2 Matrices
Exercise 5. Find transposes of the following matrices:
1. I2, I3, In
2. 03×2, O3×2, Om×n 1 −11 8 0 3 −1 0 8 14 9 3. A = 0 −2 13 7 4. B = 7 5 12 0 15 −10 0 0 6 −23 −6 4 8 0 11 0 0 0 0 17 −5 0 0 0 5. C = 0 −2 0 0 0 0 6 0 0 0 0 7
Exercise 6. Find the unknown entries of the following matrices: 0 8 −3 1. A = x −1 2 is a symmetric matrix, y z 11 3 x 2 y 2. B = −1 9 0 11 is a skew-symmetric matrix, z t −3 6 20 u v 8
Exercise 7. Find A + B and A − B of the following matrices: h i h i lOMoAR cPSD| 59085392 1. A = 1 0 −3 9 10 8 and B = 0 4 −3 −6 0 −20 5 −2 2. A = 23 and 9 0 3. and
Exercise 8. Find the unknown entries of the following matrices 1. 2. Exercise 9. Let and
. Compute AB,BA,AC,CA,BC and
CB whenever they are well-defined.
Exercise 10. Compute the products 1. Exercise 11. Compute 1.
Exercise 12. Find 2 × 2 matrices such that
1. A2 = −I, A having only real entries
2. B2 = 0, although B 6= 02
3. CD = −DC, where CD 6= 02
4. EF = 0, in which no entries of E and F are zero. lOMoAR cPSD| 59085392
Exercise 13. Which of the following matrices equal (A + B)2?
A2 + 2AB + B2, A(A + B) + B(A + B), (A + B)(B + A), A2 + AB + BA + B2.
Exercise 14. Find A2,A3; B2,B3 and C2,C3. What are Ak,Bk and Ck? and C = AB. Exercise 15. Find 1.
Exercise 16. Let p(x) = x2 − 5x + 3. Compute p(A), where 1.
Exercise 17. Prove that 2 0 0 1. A = 0 2
0 is a solution of p(x) = x3 − 3x2 + 4 0 0 −1 2.
) is a solution of p(x) = x2 − (a + d)x + (ad − bc)
Exercise 18. Let A,B ∈ Mn(K) satisfy AB = BA. Prove
1. (A + B)2 = A2 + 2AB + B2
2. A2 − B2 = (A + B)(A − B) 3. , where n ∈ N\{0} lOMoAR cPSD| 59085392 2 DETERMINANTS Exercise 19. 1. 3. 4. 2. Exercise 20. 1. 3. 2. 4.
Exercise 21. Prove the following equalities: 1. 2. 3. 4. lOMoAR cPSD| 59085392
Exercise 22. Compute the following determinants 1 1 1 1 −1 1 1 1.2.3.4. 1 1 −1 1 1 1 1
Exercise 23. Compute the following determinants of order n: 1. (order n ≥ 2) 2. (order n ≥ 2) 6. (order n+1≥3)
Exercise 24. Evaluate the following determinants: 3. (order n ≥ 2) 1.
Exercise 25. Find x satisfying: 1. 4. (order n ≥ 2) 5.(order n+1≥3) ... ... ... ... a1 7. (order n+1≥3) 8. (order n≥2) lOMoAR cPSD| 59085392 x y ... ... ... ... a2 a1 x2 9.(order n≥3)
12.a2(order n≥2) 0 0 0 ... ... ... ... 0 0 0 a2 x a2 10.(order n≥2) ... ... ... ... 2. x a1 2.
11.a2(order n + 1≥2)
Exercise 26. Solve the following systems by using Cramer’s rule
23xx11 −+ 4 xx2 2− − x 23 x= 43 = 11
x1 + 2x2 + 3x3 − 2x4 = 6 1.
2x1 − x2 − 2x3 − 3x4 = 4
3x1 − 2x2 + 4x3 = 11 3.
3x1 + 2x2 − x3 + 2x4 = 4
2x1 − 3x2 + 2x3 + x4 = −8
x1 + x2 + 2x3 = −1 2.
2x1 − x2 + 2x3 = −4
4x1 + x2 + 4x3 = −2
Exercise 27. Find the inverse (if exists) of the following matrices 0 3 1 3 2 1 2 0 1 1 1 0 0 1 1. 2 1 1 3. 2 1 3 1 1 2 0 7. 0 1 1 0 lOMoAR cPSD| 59085392 5. 3 2 2 3 2 1 0 1 1 2 0 0 1 1 2 0 1 1 1 0 0 1 2 1 0 2 0 1 2 1 3 5 2 2 1 0 6. 2. 1 1 0 4. 5 0 1 0 2 2 1 2 0 1 3 1 0 1 0 2 2
Exercise 28. Find the ranks of the following matrices 1 2 3 4 5 0 3 2 0 1 1. A = 0 1 2 3 4 3. C = 0 0 0 1 2 5. 2 4 6 8 10 0 0 0 0 0 1 2 3 4 5 1 0 0 2. B = 2 0 4 5 6 4. D = 2 3 4 3 0 0 6 7 3 0 5
Exercise 29. Find the ranks of the following matrices 2 5 7 3 5 8 1 2 3 5 7 1 2 3 1 2 3 2 5 7 12 17 1. A = 3 7 11 5 8 12 0 1 2 3 4 2. B = 4 9 14 6 10 15 1 3 5 8 11 5 11 17 7 12 18 2 4 7 11 15 3 7 11 18 25 lOMoAR cPSD| 59085392
Exercise 30. Find the ranks of the following matrices −1 2 1 1 2 3 1 3 5 1. 2 λ −2 2. 3 6 − λ 9 3. 4 12 λ + 5 3 −6 −3 4 8 − λ 12 5 15 λ + 10 3 VECTOR SPACES
Exercise 31. In R3, give two vectors u1 = (1,−2,3) and u2 = (0,1,−3).
1. Is u = (2,4,5) a linear combination of u1 and u2?
2. Find m so that v = (1,m,−3) is a linear combination of u1 and u2.
Exercise 32. Are (2,11,18),(2,−1,4),(0,5,3),(1,−1,0) linearly dependent?
Exercise 33. Are (2,−1,4),(0,5,3),(1,−1,0) linearly dependent?
Exercise 34. Show that (2,5,−1,1),(1,4,0,2),(6,15,−3,3) are linearly dependent.
Exercise 35. Show that (1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0) are linearly independent. Exercise 36. In P3[x], given
u1 = 3x2 − 2x + 1, u2 = x2 − x + 2, u3 = x3 + x + 1.
Prove that u = 2x3 + 2x2 − 5x + 4 is a linear combination of {u1,u2,u3}. Exercise 37. In P2[x], given
u1 = x2 + 3x + 1, u2 = −2x2 − x − 1, u3 = −2x2 + x + m.
Find m so that u3 is a linear combination of {u1,u2}.
Exercise 38. In P3[x], given
u1 = x3 + 2x2 + x + 1, u2 = 2x3 + x2 − x + 1, u3 = 3x3 + 3x2 − x + 2.
When u = ax3 + bx2 + cx + d is a linear combination of {u1,u2,u3}.
Exercise 39. In R4 given
u1 = (1,1,1,1), u2 = (2,3,−1,0), u3 = (−1,−1,1,1), u4 = (1,2,1,−1).
When u = (x1,x2,x3,x4) is a linear combination of
1. {u1,u2,u3}?
2. {u1,u2,u3,u4}? lOMoAR cPSD| 59085392
Exercise 40. In R3, determine if the following sets of vectors are linearly independent or dependent:
1. {u1 = (1,0,1), u2 = (1,2,3), u3 = (10,11,12), u4 = (4,5,6)}
2. {u1 = (1,0,1), u2 = (1,2,3), u3 = (2,2,4)}
3. {u1 = (1,0,1), u2 = (1,2,3), u3 = (2,2,5)}
4. {u1 = (1,0,1), u2 = (1,2,3)}
Exercise 41. In P3[x], determine if the following sets of vectors are linearly independent or dependent:
1. {u1 = x3 − 2x + 3, u2 = x2 + 1}
2. {u1 = x3 − 2x + 3, u2 = x2 + 1, u3 = 2x3 + x2 − 4x + 10}
3. {u1 = x3 − 2x + 3, u2 = x2 + 2x − 1, u3 = x3 + x2 + 2}
4. {u1 = x3, u2 = 2x2, u3 = 3x, u4 = 2x2 + 3x, u5 = 1}
Exercise 42. In M3(R), prove that the following set of vectors is linearly independent:
Exercise 43. Which of the following sets of vectors are bases of R3:
1. S1 = {(1,2,3),(0,2,3)}
3. S3 = {(1,1,2),(1,2,5),(5,3,4)}
2. S2 = {(1,2,3),(0,2,3),(0,0,5)} 4. S4 = {(−1,0,0),(−1,1,0),(1,−1,1),(2,0,5)}
Exercise 44. In R3, prove that S = {u1,u2,u3} is a basis and find the coordinates of u with respect to S:
1. u1 = (1,1,1); u2 = (1,1,2); u3 = (1,2,3); u = (6,9,14)
2. u1 = (2,1,−3); u2 = (3,2,−5); u3 = (1,−1,1); u = (6,2,−7)
3. u1 = (1,−1,0); u2 = (1,0,−1); u3 = (2,0,0); u = (6,9,14)
Exercise 45. In R3 given two sets
S1 = {(1,1,0),(0,1,1),(1,0,1)};
S2 = {(0,0,1),(1,−1,0),(1,1,1)}.
1. Prove that S1,S2 are bases of R3.
2. Find basis transformation matrices from S1 to S2 and from S2 to S1.
3. Find the coordinates of x = (1,−1,1) with respect to S1 and S2.
Exercise 46. In Rn, let
B = {(1,1,1),(1,1,2),(1,2,3)} and B0 = {(2,1,−1),(3,2,−5),(1,−1,m)}. lOMoAR cPSD| 59085392
1. Prove that B is a basis of R3.
2. Find m so that B0 is a basis of R3.
3. In the case B0 is a basis, find the transformation matrix from B into B0 and find the coordinates of u = (1,0,0)
with respect to these two bases.
Exercise 47. In Pn[x], let
B = {1,x,x2,...,xn} and B0 = {1,x − a,(x − a)2,...,(x − a)n}, where a ∈ R.
1. Prove that B and B0 are bases of Pn[x].
2. Find the transformation matrix from B into B0.
3. Find the coordinates of f(x) = a0 + a1x + a2x2 + ... + anxn with respect to B and B0.
Exercise 48. Find basis and dimension of the linear subspaces S of Rn which is spanned by
1. S = {(1,4,−1,3),(2,1,−3,−1),(0,2,1,−5)}
2. S = {(1,−4,−2,1),(1,−3,−1,2),(3,−8,−2,7)}
Exercise 49. Find basis and dimension of the linear subspaces S of M2(R) which is spanned by
Exercise 50. Find basis and dimension of the linear subspaces S of P2[x] which is spanned by
{x3 + 2x2 − 2x + 1,x3 + 3x2 − x + 4,2x3 + x2 − 7x − 7}
Exercise 51. Find basis and dimension of the null space of the following matrices: 1 3 2 1 −2 7 3. 1. 1 5 1 2. 2 3 −2 3 5 8 2 −1 1
Exercise 52. Find basis and dimension of the solution space of the following homogeneous systems of linear equations:
x + 2y − 2z + 2s − t = 0
x + 2y − z + 3s − 4t = 0 1.
x + 2y − z + 3s − 2t = 0 2.
2x + 4y − 2z − s + 5t = 0
2x + 4y − 7z + s + t = 0
2x + 4y − 2z + 4s − 2t = 0 lOMoAR cPSD| 59085392
Exercise 53. In R4, let the following subspaces
W1 = {(a,b,c,d)| b − 2c + d = 0}, W2 = {(a,b,c,d)| a = d,b = 2c}.
Find bases and dimensions of W1,W2 and W1 ∩ W2.
Exercise 54. In R5, let
W1 = h(1,3,−2,2,3),(1,4,−3,4,2),(2,3,−1,−2,9)i W2 =
h(1,3,0,2,1),(1,5,−6,6,3),(2,5,3,2,1)i.
1. Find a basis and dimension of W1 + W2
2. Find a basis and dimension of W1 ∩ W2
Exercise 55. In R4, find linear subspaces created by the following sets
(a) S1 = {u1 = (1,1,0,0),u2 = (0,1,1,0),u3 = (0,0,1,1)}
(b) S2 = {u1 = (1,1,1,0),u2 = (0,1,1,1)}
(c) S2 = {u1 = (1,2,1,2),u2 = (2,1,2,1),u3 = (3,3,3,3)}
4 Linear transformation
Exercise 56. In Rn, let i ∈ {1,...,n}. Prove that the mapping f : Rn → R defined by
f(x1,..,xi,. .,xn) = xi is a linear transformation.
Exercise 57. Let g : Pn[x] → Pn−1[x] be defined by
g(p) = p0,
p ∈ Pn[x].
Prove that g is a linear transformation
Exercise 58. Determine which of the following mappings are linear.
1. f : R3 → R3, f(x1,x2,x3) = (2x1 − x2,x3 − x2,x1)
2. f : R3 → R3, f(x1,x2,x3) = (x1,x2 + 3,x3 − x1)
3. f : R3 → R3, f(x1,x2,x3) = (x2 + x3,2x3 + x1,2x1 + x2)
4. f : R3 → R3, f(x1,x2,x3) = (x21,x22,x23) lOMoAR cPSD| 59085392
5. f : R3 → R3, f(x1,x2,x3) = (x1,x2,4)
Exercise 59. Let f : R2 → R2 be a linear transformation defined by f(3,1) = (2,−4) and f(1,1) = (0,2). Find f(x1,x2).
Exercise 60. Let f : R3 → R2 be a linear transformation defined by f(1,2,3) = (1,0),f(2,5,3) = (1,0) and f(1,0,10) = (0,1).
Find f(x1,x2,x3).
Exercise 61. Find the linear transformation f : P2[x] → P2[x] satisfying
f(1) = 1 + x2, f(x) = 3 − x2, f(x2) = 4 + 2x − 3x2.
Exercise 62. Find kernels and images of the linear transformations in Exercises ??, ??, ??, ?? and ??. Exercise 63.
Let f : R4 → R3 be a linear transformation defined by
f(x1,x2,x3,x4) = (x1 − x2 + x3,2x1 + x4,2x2 + x3 − x4).
Find a basis for Kerf and a basis for Imf. Is f injective or surjective? Exercise 64. Let f : R4 → R4 be a linear
transformation defined by f(x1,x2,x3,x4) = (x1 +2x2 +4x3 −3x4,3x1 +5x2 +6x3 −4x4,4x1 +5x2 −2x3 +3x4,3x1 +8x2 +24x3 −19x4).
Find a basis for Kerf and a basis for Imf.
Exercise 65. Let f : R4 → R3 be a mapping defined by
f(x1,x2,x3,x4) = (x1 − x2 + x3,2x1 + x4,2x2 + x3 − x4)
1. Prove that f is a linear transformation.
2. Find the transformation matrix of f with respect to the following bases
B = {u1 = (1,1,1,1),u2 = (0,1,1,1),u3 = (0,0,1,1),u4 = (0,0,0,1)},
B0 = {v1 = (1,1,1),v2 = (1,1,0),v3 = (1,0,0).}
Exercise 66. Let f : R3 → R3 be a linear transformation defined by
f(x1,x2,x3) = (2x2 + x3,x1 − 4x2,3x1).
Find the transformation matrix of f with respect to the basis
B = {e1 = (1,1,1),e2 = (1,1,0),e3 = (1,0,0)}
Exercise 67. Let f : R3 → R2 and g : R3 → R2 defined by
f(x,y,z) = (2x,y + z),
g(x,y,z) = (x − y,z).
Find f + g, 3f, 2f − 5g.
Exercise 68. Let f : R3 → R2 and g : R3 → R2 defined by lOMoAR cPSD| 59085392
f(x,y,z) = (2x,y + z),
g(x,y,z) = (y,z). Find gf.
Exercise 69. Let f and g be linear transformations on R2 defined by
f(x,y) = (y,x),
g(x,y) = (0,x).
Find fg,gf,f2,g2.
Exercise 70. Prove the following linear transformation is surjective and then find f−1.
1. f(x,y,z) = (x − 3y − 2z,y − 4z,z)
2. f(x,y,z) = (x + z,x − z,y).
Exercise 71. The linear transformation f : R4 → R4 whose transformation matrix w.r.t. B = {e1,e2,e3,e4} is 1 3 2 1 2 5 11 2 0 −1 3 1 1 2 1 3 1. Find Kerf,Imf
2. Find the transformation matrix of f w.r.t. basis B0 = {u1,u2,u3,u4}, where u for i = 1,2,3,4.
Exercise 72. Find eigenvectors and eigenvalues of the linear transformation f : R3 → R3, where the transfor-
mation matrix w.r.t. B = {e1,e2,e3} 2 −1 2 4 −5 2 1 −3 4 1. 5 −3 3 3. 5 −7 3 5. 4 −7 8 −1 0 −2 6 −9 4 6 −7 −7 1 0 1 −3 3 7 −12 6 0 lOMoAR cPSD| 59085392 2. − 4 4 4 4. −2 −6 13 6. 10 −19 10 −2 1 2 −1 −4 8 12 −24 13 5 Euclidean Spaces
Exercise 73. Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle?
1. P(3,−2,−3),Q(7,0,1),R(1,2,1)
2. P(2,−1,0),Q(4,1,1),R(4,−5,4).
Exercise 74. Determine whether the points lie on straight line.
1. A(2,4,2),B(3,7,−2),C(1,3,3)
2. D(0,−5,5),E(1,−2,4),F(3,4,2)
Exercise 75. Find a + b,2a + 3b,kak ,ka − bk
1. a = (2,−4,2),b = (3,7,−2),
2. a(1,−2,5),b(1,−2,4).
Exercise 76. Find a vector that has the same direction as v = (−2,4,2) but has length 6.
Exercise 77. Find parametric equations for the line
1. The line through the point (6,−5,2) and parallel to the vector (1,2,3).
2. The line through the point (0,14,−10) and parallel to the line x = −1 + 2t,y = 6 − 3t,z = 3 + 9t.
3. The line through the point (1,0,6) and perpendicular to the plane x + 3y + z = 5.
Exercise 78. Find an equation of the plane.
1. The plane through the point (6,3,2) and perpendicular to the vector (−2,1,5),
2. The plane through the point (4,0,−3) and with normal vector (0,1,2),
3. The plane through the point −2,8,10) and perpendicular to the line x = 1 + t,y = 2t,z = 4 − 3t,
4. The plane through the origin and parallel to the plane 2x − y + 3z = 1
Exercise 79. Which of the points P(6,2,3), Q(−5,−1,4), and R(0,3,8) is closest to the xz-plane? Which point lies in the yz-plane?
Exercise 80. What are the projections of the point (2,3,5) on the xy-, xz-, and yz-planes? Draw a rectangular box
with the origin and (2,3,5) as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices
of the box. Find the length of the diagonal of the box.
Exercise 81. Describe and sketch the surface in R3 represented by the equation x + y = 2.
Exercise 82. What does the equation x = 4 represent in R2? What does it represent in R3? Illustrate with sketches. lOMoAR cPSD| 59085392
Exercise 83. What does the equation y = 3 represent in R3? What does z = 5 represent? What does the pair of
equations y = 3, z = 5 represent? In other words, describe the set of points (x,y,z) such that y = 3 and z = 5. Illustrate with a sketch.
Exercise 84. Let P(3,2,3), Q(7,0,1) and R(1,2,1). Find the lengths of the sides of the triangle PQR. Is it a right
triangle? Is it an isosceles triangle?
Exercise 85. Find the distance from (3,7,−5) to each of the following. (i) The xy-plane (iii) The yz-plane (v) The y-axis (ii) The xz-plane
(iv) The x-axis (vi) The z-axis
Exercise 86. Find an equation of the sphere with center (1,−4,3) and radius 5. What is the intersection of this
sphere with the xz-plane?
Exercise 87. Find an equation of the sphere that passes through the point (4,3,−1) and has center (3,8,1).
Exercise 88. Show that the equation represents a sphere, and find its center and radius.
(i) x2 + y2 + z2 − 6x + 4y − 2z = 11
(ii) x2 + y2 + z2 + 8x − 6y + 2z + 17 = 0
(iii) 2x2 + 2y2 + 2z2 = 8x − 24z + 1
(iv) 4x2 + 4y2 + 4z2 − 8x + 16y = 1
Exercise 89. Prove that the midpoint of the line segment from P1(x1,y1,z1) to P2(x2,y2,z2) is .
Exercise 90. Find the lengths of the medians of the triangle with vertices
A(1,2,3), B(−2,0,5), C(4,1,5).
Exercise 91. Find an equation of a sphere if one of its diameters has endpoints (2,1,4) and (4,3,10).