- Exercises
Write in
the vector
-
as a linear combination of the vectors
-
Write in
the vector
-
as a
linear combinationMDLD/linear combination
of the vectors
-
===Exercise Exercise 23.3
change===
Let
be a
field,MDLD/field
and let
be a
-vector space.MDLD/vector space
Show that the following statements hold.
- For a family
,
,
of elements in
,
linear spanMDLD/linear span
is a
linear subspaceMDLD/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,MDLD/field
and let
be a
-vector space.MDLD/vector space
Let
,
,
be a family of vectors in
and
,
,
another family of vectors in
. Then, for the
spanned linear subspaces,MDLD/spanned linear subspaces
the inclusion
holds, if and only if
holds for all
.
Let
be a
field,MDLD/field
and let
be a
-vector space.MDLD/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 subspacesMDLD/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.MDLD/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.
===Exercise Exercise 23.10
change===
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
-
===Exercise Exercise 23.18
change===
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 ringMDLD/polynomial ring (field 1)
over
. Let
.
Show that the set of all polynomials of degree
is a
finite dimensionalMDLD/finite dimensional
subspaceMDLD/subspace
of
. What is its
dimension?MDLD/dimension (vs)
Show that the set of all real
polynomialsMDLD/polynomials (field 1)
of
degreeMDLD/degree (polynomial)
, which have a zero for
and for
, form a
finite-dimensionalMDLD/finite-dimensional
linear subspaceMDLD/linear subspace
in
. Determine its
dimension.MDLD/dimension (vs)
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 fieldMDLD/finite field
with
elements, and let
be an
-dimensionalMDLD/dimensional (vs)
vector space.MDLD/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 systemMDLD/generating system (vs)
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 subspacesMDLD/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
polynomialsMDLD/polynomials (field 1)
of
degreeMDLD/degree (Polynom)
, which have a zero at
, at
and at
, is a
finite dimensionalMDLD/finite dimensional
subspaceMDLD/subspace (linear)
of
. Determine the
dimensionMDLD/dimension (vs)
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 subspaceMDLD/linear subspace
they span. Prove that the family is linearly independent if and only if the dimension of
is exactly
.