1. Calculate the determinant . 2. Calculate the minor of the element of the matrix: .

Analog tests on algebra

1. Calculate the determinant . 2. Calculate the minor of the element of the matrix: .

3. Find the matrix if . 4. Calculate . 5. Find if .

6. Solve the system and find . 7. Find the length of the vector .

8. Find the scalar product of the vectors and .

9. Find the angle between vectors and .

10. What value of x are the vectors and linearly dependent for?

11. What value of x are the vectors and orthogonal for?

12. Find the true equality for every matrix…. or …

13. If you replace any two rows in a determinant of the third order then … 14. If you add one row in a determinant to another row then …

15. Let . Find . 16. Let . Find the transposed matrix.

17. Vectors and are such that and the angle between these vectors is equal to . Find the scalar product .

18. Calculate the determinant . 19. Calculate the minor of the element of the matrix: .

20. Find the matrix if . 21. Calculate . 22. Find if .

23. Solve the system and find . 24. Find the length of the vector .

25. Find the scalar product of two vectors and .

26. Find the angle between vectors and .

27. What value of y are the vectors and linearly dependent at?

28. What value of y are the vectors and orthogonal at?

29. Calculate the determinant . 30. Calculate the determinant . 31. Let . Find .

32. Let . Find . 33. If a determinant has two columns of which the elements are proportional respectively, then…

34. What value is the determinant of identity matrix equal to?

35. Let , . Find . 36. Let , . Find .

37. Let . Find . 38. Calculate the determinant:

39. Calculate the minor of the element of the matrix: . 40. Find the matrix if .

41. Calculate . 42. Find if . 43. Solve the system and find . 44. Find the length of the vector . 45. Find the scalar product of the vectors and .

46. Find the angle between the vectors and .

47. What value x are the vectors and linearly dependent for?

48. What value x are the vectors and orthogonal for?

49. Let . Find the minor . 50. Let the determinant be given. Find the cofactor .

51. Find the equality that is true for every matrix… or …

52. Let . Find . 53. Let . Find . 54. Calculate the determinant: .

55. Calculate the cofactor of the element of the matrix: .

56. Find the matrix if . 57. Calculate .

58. Find the inverse matrix if . 59. Solve the system of equations and find . 60. Find . 61. Which of the following sets is a linear space if the operations of addition and multiplication on a real number for elements of the set are determined by the ordinary way? … the set of polynomials £ 2… the set of polynomials ³ 2…

62. Find the angle between the vectors and .

63. What value x are the vectors and linearly dependent for?

64. What value x are the vectors and orthogonal for? 65. Solve the equation

66. Let . Find . 67. Let . Find .

68. The angle between two vectors and is found by the following formula:

69. Find if and .

70. Find the scalar product of the vectors and .

71. The determinant of the matrix is called the number which is equal to:

72. The value of the determinant of a square matrix of the third order after its transposing:

73. Calculate the determinant . 74. Calculate the determinant .

75. If the elements of two rows are proportional respectively then the determinant: 76. The rank of a matrix is called:

77. Find the rank of the matrix . 78. Find the inverse matrix for the matrix :

79. Solve the matrix equation . 80. Find the product of the matrices × .

81. Find if . 82. Solve the system and find .

83. Find the rank of the matrix .

84. The scalar product of non-zero vectors and is equal to zero if and only if ….

85. For vectors and the formula defines:

86. Find if and the angle between and is equal to :

87. For vectors and the formula defines:

88. Find the angle between vectors and .

89. Find the angle between vectors and .

90. Which conditions do the vectors and satisfy to?

91. Which conditions do the vectors and satisfy to?

92. Find the scalar product of vectors and .

93. Calculate the determinant . 94. Calculate the determinant .

95. Calculate if and

96. Calculate if and . 97. Calculate if .

98. Solve the system of linear equations 99. Solve the system of linear equations

100. Find if ,

101. Calculate the determinant: 102. Calculate the cofactor of the element of the matrix .

103. Find the matrix if . 104. Calculate if and

105. Calculate if and . 106. Calculate if .

107. Find the module of the complex number . 108. Find the sum of and .

109. Find the product of and . 110. Find the complex number which is equal to .

111. Let . Find . 112. Find the trigonometric form of the complex number . 113. Calculate . 114. Find . 115. Find the complex number which is equal to . 116. Let . Find .

117. Let be vectors of a linear space R. The vectors are linearly dependent if

118. A basis of an n -dimensional linear space R is

119. Let be a basis of a linear space. Find the coordinates of the vector in this basis.

120. Let be a basis of a linear space. Find the coordinates of the vector in this basis.

121. Let be a basis of a linear space. Find the coordinates of the vector in this basis.

122. Find characteristic numbers of the matrix . 123. Find eigenvectors of the matrix .

124. A non-zero element x of a linear space R is an eigenvector of a linear transformation A if

125. Find characteristic numbers of the matrix . 126. Find eigenvectors of the matrix .

127. Let A be a linear transformation of a linear space R. Then trivial invariant subspaces of the linear transformation A are

128. Find characteristic numbers of a linear transformation with the matrix .

129. Find eigenvectors of a linear transformation with the matrix .

130. Find characteristic numbers of a linear transformation with the matrix .

131. Find eigenvectors of a linear transformation with the matrix .

132. Let A be a matrix of dimension (3; 6). Then r(A) (the rank of the matrix A) can be the following:

133. Let . Find the determinant of A.

134. Reduce the quadratic form to canonical form:

135. Reduce the quadratic form to canonical form:

136. Find the least natural value of the parameter a for which the quadratic form is positive definite.

137. Find the greatest integer value of the parameter b for which the quadratic form is negative definite.

138. Find all values of the parameter c for which the quadratic form is definite.

139. A quadratic form is called quasi-definite if …

140. Find the least natural value of the parameter b for which the quadratic form is positive definite.

141. Find the greatest integer value of the parameter a for which the quadratic form is negative definite.

142. Let a vector be given. Express this vector through the new basis if , .

143. Let and be subspaces of a linear space , . Then … ….

144. The linear hull of four vectors is

145. Determine the dimension of the subspace of solutions of the system .

146. Let be old and new bases respectively in a 2-dimensional linear space, and let , . Then the transition matrix from the old basis to new is

147. Let a vector be given. Express this vector through the new basis if , .

148. Determine the dimension of the subspace of solutions of the system .

149. Determine the dimension of the subspace of solutions of the system .

150. Let and be subspaces of a linear space , . If , , then is equal to

151. Let be old and new bases respectively in a 3-dimensional linear space, and let , , . Then the transition matrix from the old basis to new is

152. Find the matrix of the following linear transformation in a 2-dimensional linear space.

153. Let A be a linear transformation in a 3-dimensional linear space. Write this transformation in the coordinate form if

154. The transformation A is turning each vector of the plane xOy on angle . Find the matrix of the transformation

155. Let A and B be the following linear transformations respectively: and Find

156. The transformation A is turning every vector of the plane xOy on angle . Find the matrix of the transformation

157. Let A be the following linear transformation: . Find the matrix of the inverse linear transformation.

158. A linear transformation A is given in a linear space with the basis Find the matrix of the inverse transformation if

159. Let A be a linear transformation in a 2-dimensional linear space. Write this transformation in the coordinate form if

160. Let the following linear transformations be given: and Find the transformation

161. Let A be a linear transformation in a linear space with the basis Find the matrix if

162. Let the following linear transformations be given: and . Find the transformation

163. Let A be the following linear transformation: Find the matrix of the inverse linear transformation. 164. Find the rank of the matrix .

165. Find the rank of the matrix . 166. Find the rank of the matrix .

167. Find the rank of the matrix . 168. Find the rank of the matrix .

169. Find the rank of the matrix . 170. Find the rank of the matrix .

171. Find the rank of the matrix . 172. Find the rank of the matrix .

173. Find the rank of the matrix . 174. Normalize the vector .

175. Normalize the vector . 176. Find the length of the vector .

177. Find the length of the vector .

178. What positive value do the vectors and have the same lengths for?

179. What negative value do the vectors and have the same lengths for?

180. Determine the angle between vectors and .

181. Find the least value of such that a transformation A defined be the equality is orthogonal.

182. Find normalized vectors in a 2-dimensional Euclidean space with the basis that are orthogonal to the vector .

183. A linear transformation A of a Euclidean space is called orthogonal if …

184. A transformation A of a linear space V is linear if for every and for every real number the following holds:

185. A group G is abelian if for every the following holds:

186. An element of a group has infinite order if …

187. Find the module of the complex number . 188. Find if and .

189. Find if and . 190. Find the algebraic form of the complex number

191. Let . Find . 192. Find the trigonometric form of the complex number . 193. Calculate .

194. Find . 195. Find characteristic numbers of a linear transformation with the matrix .

196. Calculate the determinant . 197. Calculate the determinant .

198. Find the rank of the matrix .

199. Let A be a matrix of dimension (4; 8). Then r(A) (the rank of the matrix A) can be the following:

200. Calculate the determinant .

Дата добавления: 2015-11-04