Algebra
.docxAlgebra and geometry
Вопрос № 1
1 Let
,
be vectors in
.
Find the angle between these vectors.
2 0;
Вопрос № 2
Let
,
be vectors in
.
Find the angle between these vectors .
2
;
Вопрос № 3
Let
,
be vector in
.
Find the angle between these vectors.
2
;
Вопрос № 4
Let
,
be vectors in
.
Find the angle between these vectors
0;
Вопрос № 5
Let
,
be vectors in
.
Find the angle between these vectors.
0;
Вопрос № 6
Let
,
be vectors in
.
Find the angle between these vectors.
0;
Вопрос № 7
Let
,
be vectors in
.
Find the angle between these vectors..
;
Вопрос № 8
Let
,
be vectors in
.
Find the angle between these vectors.
;
.
Вопрос № 9
9) Let
and
be vectors in
such that
.
Find
if:
.
2;
Вопрос № 10
Let
and
be vectors in
such that
.
Find
if:
.
-1;
Вопрос № 11
$$1 Let
and
be vectors in
such that
.
Find
if:
.
$$2 12;
Вопрос № 12
$$1 Let
and
be vectors in
such that
.
Find
if:
.
$$2
;
Вопрос № 13
$$1 Let
and
be vectors in
such that
.
Find
if:
.
$$2
;
Вопрос № 14
$$1 Let
and
be vectors in
such that
.
Find the angle between
and
if:
.
$$2
;
Вопрос № 15
$$1 Let
and
be vectors in
such that
.
Find the angle between
and
if:
.
$$2
;
Вопрос № 16
$$1 Determine
so that vectors
and
are orthogonal, where
.
$$2
;
Вопрос № 17
$$1 Determine
so that vectors
and
are orthogonal, where
.
$$2 -6
Вопрос № 18
$$1 Determine
so that vectors
and
are orthogonal, where
.
$$2
Вопрос № 19
$$1 Determine
so that vectors
and
are orthogonal, where
.
$$2
Вопрос № 20
$$1 Determine
so that vectors
and
are orthogonal, where
.
$$2
Вопрос № 21
$$1 Find
and
if
.
$$2
Вопрос № 22
$$1 Find
and
if
.
$$2
Вопрос № 23
$$1 Find
and
if
.
$$2
Вопрос № 24
$$1 Find
and
if
.
$$2
Вопрос № 25
$$1 Find
and
if
.
$$2
Вопрос № 26
$$1 Solve the system
.
$$2
Вопрос № 27
$$1 Solve the system
.
$$2
Вопрос № 28
$$1 Solve the system
.
$$2 No solution;
Вопрос № 29
$$1 Solve the system
.
$$2
Вопрос № 30
$$1 Solve the system
.
$$2 No solution;
Вопрос № 31
$$1 Solve the system
.
$$2
Вопрос № 32
$$1 Determine
so that the system
has the unique solution.
$$2
Вопрос № 33
$$1 Determine
so that the system
has the unique solution.
$$2
Вопрос № 34
$$1 Determine
so that the system
has the unique solution.
$$2
Вопрос № 35
$$1 Determine
so that the system
has the unique solution.
$$2
Вопрос № 36
$$1 Determine
so that the system
has the unique solution.
$$2
Вопрос № 37
$$1 Determine
so that the system
has more the one solution.
$$2
Вопрос № 38
$$1 Determine
so that the system
has more then one solution.
$$2
Вопрос № 39
$$1 Determine
so that the system
has more then one solution.
$$2
Вопрос № 40
$$1 $$1 Determine
so that the system
has more then one solution.
$$2
Вопрос № 41
$$1 Determine
so that the system
has more then one solution.
$$2
Вопрос № 42
$$1 Let
.
Find
.
$$2
Вопрос № 43
$$1 Let
.
Find
.
$$2
Вопрос № 44
$$1 Let
.
Find
.
$$2
does not exist;
Вопрос № 45
$$1 Let
,
.
Find
$$2
Вопрос № 46
$$1 Let
,
.
Find
$$2
Вопрос № 47
$$1 Let
,
.
Find
$$2
Вопрос № 48
$$1 Given
.
Find a vector
such that
.
$$2
;
Вопрос № 49
$$1 Given
.
Find a vector
such that
.
$$2
;
.
Вопрос № 50
$$1 Given
.
Find a vector
such that
.
$$2
;
Вопрос № 51
$$1 Write the vector
as a linear combination of the vectors
.
$$2
Вопрос № 52
$$1 Write the vector
as a linear combination of the vectors
.
$$2
Вопрос № 53
$$1 Write the vector
as a linear combination of the vectors
.
$$2
Вопрос № 54
$$1 Write the vector
as a linear combination of the vectors
.
$$2
Вопрос № 55
$$1 Write
as a linear combination of matrices
.
$$2
;
Вопрос № 56
$$1 Write
as a linear combination of matrices
$$2
;
Вопрос № 57
$$1 Write
as a linear combination of matrices
$$2
;
Вопрос № 58
Write
as a linear combination of matrices
$$2
;
Вопрос № 59
$$1 Find the rank of matrix
.
$$2
;
Вопрос № 60
$$1 Find the rank of matrix
.
$$2
;
Вопрос № 61
$$1 Find the rank of matrix
.
$$2
;
Вопрос № 62
$$1 Find the rank of matrix
.
$$2
;
Вопрос № 63
$$1 Find the rank of matrix
.
$$2
;
Вопрос № 64
$$1 Find the rank of matrix
.
$$2
;
Вопрос № 65
$$1 Find the rank of matrix
.
$$2
;
Вопрос № 66
$$1 Find the rank of matrix
.
$$2
;
Вопрос № 67
$$1 Find a dimension of the
solution space
of the homogeneous system
.
$$2
Вопрос № 68
$$1 Find a dimension of the
solution space
of the homogeneous system
.
$$2
Вопрос № 69
$$1 $$1 Find a dimension of
the solution space
of the homogeneous system
.
$$2
$$1 Find a dimension of the
solution space
of the homogeneous system
.
$$2
Вопрос № 70
$$1 Find a dimension of the
solution space
of the homogeneous system
.
$$2
.
Вопрос № 71
$$1 Find a dimension of the
subspace
of
generated by vectors
.
$$2
;
Вопрос № 72
$$1 Find a dimension of the
subspace
of
generated by vectors
.
$$2
;
Вопрос № 73
$$1 Find a dimension of the
subspace
of
generated by vectors
.
$$2
;
Вопрос № 74
$$1 Find a dimension of the
subspace
of
generated by vectors
.
$$2
;
Вопрос № 75
$$1 Find a dimension of the
subspace
of
generated by vectors
,
.
$$2
;
Вопрос № 76
$$1 Let
be linear operators on
defined by
.
Find
.
$$2
;
Вопрос № 77
$$1 Let
be linear operators on
defined by
.
Find
.
$$2
;
Вопрос № 78
$$1 Let
be linear operators on
defined by
.
Find
.
$$2
;
Вопрос № 79
$$1 Let
be linear operators on
defined by
.
Find
.
$$2
;
Вопрос № 80
$$1 Let
be linear operators on
defined by
.
Find
.
$$2
;
Вопрос № 81
$$1 Let
be linear operators on
defined by
.
Find
.
$$2
;
Вопрос № 82
$$1 Let
be linear operators on
defined by
.
Find
.
$$2
;
Вопрос № 83
$$1 Which of the following
mappings
are
linear?
$$2
;
Вопрос № 84
$$1 Which of the following
mappings
are
linear?
$$2
;
Вопрос № 85
$$1 Which of the following
mappings
are
linear?
$$2
;
Вопрос № 86
$$1 Find the matrix of the
linear operator on
defined by
with respect to the usual basis.
$$2
;
Вопрос № 87
$$1 Find the matrix of the
linear operator on
defined by
with respect to the usual basis.
$$2
;
Вопрос № 88
$$1 Find the matrix of the
linear operator on
defined by
with respect to the usual basis.
$$2
;
Вопрос № 89
$$1 Find the matrix of the
linear operator on
defined by
with respect to the usual basis.
$$2
;
Вопрос № 90
$$1 Find the matrix of the
linear operator on
defined by
with respect to the usual basis.
$$2
;
Вопрос № 91
$$1 Let
be the linear operator on
defined by
.
Find
.
$$2
Вопрос № 92
$$1 Let
be the linear operator on
defined by
.
Find
.
$$2
Вопрос № 93
$$1 Let
be the linear operator on
defined by
.
Find
.
$$2
Вопрос № 94
$$1 Let
be the linear operator on
defined by
.
Find
.
$$2
Вопрос № 95
$$1 Let
be the linear operator on
defined by
.
Find
.
$$2
Вопрос № 96
$$1 Let
be the linear operator on
defined by
.
Find
.
$$2
Вопрос № 97
$$1 For the matrix
,
find the cofactor of the entry
.
$$2
Вопрос № 98
$$1 For the matrix
,
find the cofactor of the entry
.
$$2
Вопрос № 99
$$1 For the matrix
,
find the cofactor of the entry
.
$$2
Вопрос № 100
$$1 Evaluate the determinant
of the matrix
.
$$2
Вопрос № 101
$$1 Evaluate the determinant
of the matrix
.
$$2
Вопрос № 102
$$1 Evaluate the determinant
of the matrix
.
$$2
Вопрос № 103
$$1 Evaluate the determinant
of the matrix
.
$$2
Вопрос № 104
$$1 Evaluate the determinant
of the matrix
.
$$2
Вопрос № 105
$$1 Let
.
Find
.
$$2
Вопрос № 106
$$1 Let
.
Find
.
$$2
Вопрос № 107
$$1 Let
.
Find
.
$$2
Вопрос № 108
$$1 Let
.
Find
.
$$2
Вопрос № 109
$$1 Let
.
Find
.
$$2 Does not exist;
Вопрос № 110
$$1 Let
.
Find
.
$$2 Does not exist;
Вопрос № 111
$$1 Let
be the linear operator on
defined by
.Find
.
$$2
Вопрос № 112
$$1 Let
be the linear operator on
defined by
.Find
.
$$2
Вопрос № 113
$$1 Let
be the linear operator on
defined by
.
Find
.
$$2
Вопрос № 114
$$1 Let
be the linear operator on
defined by
.Find
.
$$2
Вопрос № 115
$$1 Let
be the linear operator on
defined by
.
Find
.
$$2
Вопрос № 116
$$1 Find all eigenvalues of
the matrix
.
$$2
Вопрос № 117
$$1 Find all eigenvalues of
the matrix
.
$$2
Вопрос № 118
$$1 Find all eigenvalues of
the matrix
.
$$2
Вопрос № 119
$$1 Find all eigenvalues of
the matrix
.
$$2 Does not exist;
Вопрос № 120
$$1 Find all eigenvalues of
the matrix
.
$$2
Вопрос № 121
$$1 Find the slope and the
segment cut out on the y-axis
of the straight line
.
$$2
Вопрос № 122
$$1
Find the slope and the segment cut out on the y-axis
of the straight line
.
$$2
Вопрос № 123
$$1
Find the slope and the segment cut out on the y-axis
of the straight line
.
$$2
.
Вопрос № 124
$$1 Find the equation of the
straight line that passes through the origin and is parallel to the
straight line
;
$$2
Вопрос № 125
$$1 Find the equation of the
straight line that passes through the origin and is parallel to the
straight line
;
$$2
;
Вопрос № 126
$$1
Find the equation of the straight line that passes through the point
and is parallel to the straight line
.
$$2
;
Вопрос № 127
$$1
Find the equation of the straight line that passes through the point
and is parallel to the straight line
.
$$2
;
Вопрос № 128
. $$1 Find the equation of
the straight line that passes through the point
and is perpendicular to the straight line
.
$$2
;
Вопрос № 129
$$1 Find the equation of the
straight line that passes through the point
and is perpendicular to the straight line
.
$$2
;
Вопрос № 130
$$1 Find the equation of the
straight line that passes through the point
and is perpendicular to the straight line
.
$$2
;
Вопрос № 131