Algebra_final_preparation_questions
.pdf170. |
Let |
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171. |
Let |
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172. |
Solve the equation x2 |
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173.The rank of a matrix is equal to ...
174. |
Find the product of the matrices |
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175. |
Find |
A2 if |
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176. |
Find the scalar product of vectors a e1 2e2 3e3 and b 7e1 e2 . |
177. |
Let v1 , v2 ,..., vm be vectors of a linear space R. The vectors v1 , v2 ,..., vm are linearly dependent |
if |
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there exist real numbers a1 , a2 ,..., am , not all of them 0, such that a1v1 a2v2 ... amvm 0.
there exist real numbers a1 , a2 ,..., am such that a1v1 a2v2 ... amvm 0.
a1v1 a2v2 |
... amvm 0 |
iff a1 0, a2 0,...., am 0. |
a1v1 a2v2 |
... amvm 0 |
for all real numbers a1 , a2 ,..., am . |
they are orthogonal. |
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178.A basis for n-dimensional linear space R is formed by ...
a set of n linearly independent vectors from R a set of n linearly dependent vectors from R a set of n pairwise orthogonal vectors from R a set of n arbitraryly distinct vectors from R a set of n zero vectors from R.
179.Let e1 (1;1), e2 ( 1; 0) be a basis of a linear space. Find the coordinates of the vector x (5;8) in this basis.
180. Let e1 ( 1; 2), e2 (3; 5) be a basis of a linear space. Find the coordinates of the vector x (4; 7) in this basis.
181.Let A be a matrix of size (6x3). Then rank of the matrix A can be the following:
2 3
182. Let A
0 1
1
. Find the determinant of A.
2
183.The linear span of four vectors x, y, z,u is
the set of all linear combinations of these vectors the set consisting only of these vectors
the sum of these vectors
the set of all scalar products of these vectors
the set of vectors that are orthogonal to each of these vectors
184. Determine the dimension of the subspace formed by the solutions of the system
x1 x2 x3 0 |
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x2 x3 |
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x |
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185. |
Let |
{e , e } and {e , e |
}be old and new bases respectively in a 2-dimensional linear space, |
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and let e |
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186. |
Let vector x 2e |
3e |
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be given. Find resolution of this vector in the new basis e , e if |
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e e e |
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187. |
Let |
{e , e } and {e , e |
}be old and new bases respectively in a 3-dimensional linear space, |
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and let |
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2e . Then the transition matrix |
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from the old basis to new basis is ...
188.Find the length of the vector x e1 
3 e2 
2 6e3 .
189.Let A be a matrix of size 2x7. Then rank of the matrix A can be the following:
sin cos
190. Find determinant: cos sin
191. What object will you get in the result of addition of two vectors?
Line segment Number vector matrix
none of these
192. Determinant |AB| equals to
A B
A B
A B
A2 B2
AT BT .
193.What is the dimension of the vector space of square matrices of size 2?
194. What is the rank of matrix 0 |
0 ? |
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195.What operations are defined for elements of the Vector Space?
Addition and multiplication by a number Four arithmetic operations
Only division None of these only addition
196.From expression v1 2v2 3v3 n vn we can conclude the following
System of vectors v1 , v2 , , |
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is linearly independent |
System of vectors v1 , v2 , , |
vn |
is linearly dependent |
Dependence is not defined |
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Vectors v1 , v2 , , vn are orthogonal
None of these
197.Find coordinates of vector x=(90, -10) in basis e1(2;0), e2(1, 1).
198.Find coordinates of vector x=(4, 20) in basis e1(1;3), e2(0, 2).
199.Compute
200.Compute the value of the function for x=A if 
201.Compute the product
202.Compute the product
203. |
What is true for the determinant ka |
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204. |
Find product of matrices: 2 |
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205.Which of the following matrices have inverse?
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None of these |
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206. |
Find inverse for matrix A= 1 |
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207. |
Find inverse for matrix A= 1 |
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208. |
Express the given vector x |
as a linear combination of the given vectors a |
and b : |
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x (1, 0) , a (1, 1) , |
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(0, 1) . |
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209. |
Express the given vector x |
as a linear combination of the given vectors a |
and b : |
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x (22, 11) , a (1, 1) , |
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210. |
Express the given vector x |
as a linear combination of the given vectors a |
and b : |
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x (1, 1) , a (2, 1) , |
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a b a b
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211. |
Express the given vector x |
as a linear combination of the given vectors a and b : |
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x (4, 3) , a (2, 5) , b |
( 1, 0) . |
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212. |
Verify that vectors a and b |
form basis in the plane and resolve the given vector |
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according to this basis: x (49, 14) , a (2, 5) , |
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(9, 4) . |
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213. |
Find scalar product of the given vectors a (4, 2, 4) , |
b |
(7, 3, 2) |
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214.Find | a |2 if vector a (5, 2, 5) .
215. |
Find vector x which is collinear to the vector |
a (2, 1, 1) and satisfies the |
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condition (x, a) 3 |
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216. |
Let matrices A and B are given: A= 2 |
1 , B= 7 |
3 . Find At -2B. |
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217.Compute the value of the function for x=A if
218.Compute the value of the function for x=A if
219.Find rank of the matrix
220.Find rank of the matrix