- Warm-up-exercises
Write in
the vector
-
as a linear combination of the vectors
-
Write in
the vector
-
as a
linear combination
of the vectors
-
Let
be a
field,
and let
be a
-vector space.
Let
,
,
be a family of vectors in
, and let
be another vector. Assume that the family
-
is a system of generators of
, and that
is a linear combination of the
,
.
Prove that also
,
,
is a system of generators of
.
Let
be a field and let
be a
-vector space. Prove the following facts.
- Let
,
,
be a family of subspaces of
. Prove that also the intersection
-
![{\displaystyle {}U=\bigcap _{j\in J}U_{j}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/884d8c6452e4bcc98c72604b9de56bed8bfb8e4b)
is a subspace.
- Let
,
,
be a family of elements of
and consider the subset
of
which is given by all linear combinations of these elements. Show that
is a subspace of
.
- The family
,
,
is a system of generators of
if and only if
-
![{\displaystyle {}\langle v_{i},\,i\in I\rangle =V\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afd8a63388e1a75b62696ce79ed5c172e38fa80e)
Show that the three vectors
-
in
are
linearly independent.
Give an example of three vectors in
such that each two of them is linearly independent, but all three vectors together are linearly dependent.
Let
be a field, let
be a
-vector space and let
,
,
be a family of vectors in
. Prove the following facts.
- If the family is linearly independent, then for each subset
,
also the family
,
is linearly independent.
- The empty family is linearly independent.
- If the family contains the null vector, then it is not linearly independent.
- If a vector appears several times in the family, then the family is not linearly independent.
- A vector
is linearly independent if and only if
.
- Two vectors
and
are linearly independent if and only if
is not a scalar multiple of
and vice versa.
Let
be a field, let
be a
-vector space, and let
,
be a family of vectors in
. Let
,
be a family of elements
in
. Prove that the family
,
,
is linearly independent
(a system of generators of
, a basis of
),
if and only if the same holds for the family
,
.
Determine a basis for the solution space of the linear equation
-
![{\displaystyle {}3x+4y-2z+5w=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ae6a3ab7d716d56b42166f1d1f874ab3635f7f)
Determine a basis for the solution space of the linear system of equations
-
Prove that in
, the three vectors
-
are a basis.
Establish if in
the two vectors
-
form a basis.
Let
be a field. Find a linear system of equations in three variables, whose solution space is exactly
-
- Hand-in-exercises
Write in
the vector
-
as a linear combination of the vectors
-
Prove that it cannot be expressed as a linear combination of two of the three vectors.
Establish if in
the three vectors
-
form a basis.
Establish if in
the two vectors
-
form a basis.
Let
be the
-dimensional standard vector space over
, and let
be a family of vectors. Prove that this family is a
-basis of
if and only if the same family, considered as a family in
, is a
-basis of
.
Let
be a field, and let
-
![{\displaystyle {}{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\in K^{n}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6767d116e183765bb5a1ed3d1bab08e191ab056)
be a nonzero vector. Find a linear system of equations in
variables with
equations, whose solution space is exactly
-