- 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.
Show that the following statements hold.
- For a family
,
,
of elements in
,
linear span
is a
linear subspace
of
.
- The family
,
,
is a spanning system 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)
Let
be a
field,
and let
be a
-vector space.
Let
,
,
be a family of vectors in
and
,
,
another family of vectors in
. Then, for the
spanned linear subspaces,
the inclusion
holds, if and only if
holds for all
.
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
.
We consider in
the
linear subspaces
-
![{\displaystyle {}U=\langle {\begin{pmatrix}2\\1\\4\end{pmatrix}},{\begin{pmatrix}3\\-2\\7\end{pmatrix}}\rangle \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c75efc1cab2c46d34c044a107910a0f85d18755)
and
-
![{\displaystyle {}W=\langle {\begin{pmatrix}5\\-1\\11\end{pmatrix}},{\begin{pmatrix}1\\-3\\3\end{pmatrix}}\rangle \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c07b3d73a1b0b57f2e5dcc1c9c1a003b4545a87)
Show that
.
Show that the three vectors
-
in
are
linearly independent.
Find, for the vectors
-
in
, a non-trivial representation of the zero vector.
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
-
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
-
Let
be a field, and let
be a
-vector space of dimension
.
Suppose that
vectors
in
are given. Prove that the following facts are equivalent.
form a basis for
.
form a system of generators for
.
are linearly independent.
Let
be a field, and let
denote the
polynomial ring
over
. Let
.
Show that the set of all polynomials of degree
is a
finite dimensional
subspace
of
. What is its
dimension?
Show that the set of all real
polynomials
of
degree
, which have a zero for
and for
, form a
finite-dimensional
linear subspace
in
. Determine its
dimension.
Let
be a field, and let
and
be two finite-dimensional
vector spaces with
-
![{\displaystyle {}\dim _{}{\left(V\right)}=n\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab72054f9cfabc8640a4cd323e0bb992ea9a6ceb)
and
-
![{\displaystyle {}\dim _{}{\left(W\right)}=m\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acbfb246ff654aec53181220465fda150380b620)
What is the dimension of the Cartesian product
?
Let
be a finite-dimensional vector space over the complex numbers, and let
be a basis of
. Prove that the family of vectors
-
form a basis for
, considered as a real vector space.
Let
be a
finite field
with
elements, and let
be an
-dimensional
vector space.
Let
be an enumeration
(without repetitions)
of the elements from
. After how many elements can we be sure that these form a
generating system
of
.
- 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.
We consider in
the
linear subspaces
-
![{\displaystyle {}U=\langle {\begin{pmatrix}3\\1\\-5\\2\end{pmatrix}},{\begin{pmatrix}2\\-2\\4\\-3\end{pmatrix}},{\begin{pmatrix}1\\0\\3\\2\end{pmatrix}}\rangle \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9882680841718818ed2dc524733651c40263bae7)
and
-
![{\displaystyle {}W=\langle {\begin{pmatrix}6\\-1\\2\\1\end{pmatrix}},{\begin{pmatrix}0\\-2\\-2\\-7\end{pmatrix}},{\begin{pmatrix}9\\2\\-1\\10\end{pmatrix}}\rangle \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/094488c9ecc8e42fb4226eb753f89dfa75593429)
Show that
.
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
.
Show that the set of all real
polynomials
of
degree
, which have a zero at
, at
and at
, is a
finite dimensional
subspace
of
. Determine the
dimension
of this vector space.
Let
be a field, and let
be a
-vector space. Let
be a family of vectors in
, and let
-
![{\displaystyle {}U=\langle v_{i},\,i=1,\ldots ,m\rangle \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0f805882675ea82f51f9f1ddf91f57a1a2b63c)
be the
linear subspace
they span. Prove that the family is linearly independent if and only if the dimension of
is exactly
.