Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 8/refcontrol
- Exercise for the break
Determine the dimensionMDLD/dimension (vs) of the space of all -matrices.MDLD/matrices
- Exercises
Determine the dimensionMDLD/dimension (vs) of the solution space of the linear systemMDLD/linear system
and
in the variables .
Let be a field,MDLD/field and . Show that the set of all diagonal matricesMDLD/diagonal matrices is a linear subspaceMDLD/linear subspace in the space of all -matricesMDLD/matrices over , and determine its dimension.MDLD/dimension (vs)
Show that the set of all symmetricMDLD/symmetric (matrix) -matricesMDLD/matrices is a linear subspaceMDLD/linear subspace in the space of all -matrices, and determine its dimension.MDLD/dimension (vs)
Let be a field,MDLD/field and . Show that the set of all upper triangular matricesMDLD/upper triangular matrices is a linear subspaceMDLD/linear subspace in the space of all -matricesMDLD/matrices over , and determine its dimension.MDLD/dimension (vs)
===Exercise Exercise 8.6
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,MDLD/field and let be a -vector spaceMDLD/vector space of finite dimension.MDLD/dimension (fgvs) Let denote a linear subspaceMDLD/linear subspace with . Show that holds.
Let be real numbers. We consider the three vectors
Give examples for such that the linear subspace generated by these vectors has dimension .
Let be a field, and let and be two finite-dimensional vector spaces with
and
What is the dimension of the Cartesian product ?
Let be an -dimensional -vector spaceMDLD/vector space ( a field), and let be linear subspacesMDLD/linear subspaces of dimensionMDLD/dimension (vs) and . Suppose that holds. Show that .
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 (vs) linear subspaceMDLD/linear 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 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.
Consider the standard basis in and the three vectors
Prove that these vectors are linearly independent, and extend them to a basis by adding an appropriate standard vector as shown in the base exchange theorem. Can one take any standard vector?
We consider the linear equationsMDLD/linear equations
over .
- Determine a basisMDLD/basis (vs) of the solution space of this linear system.
- Complete the basis to a basis of the solution space of the linear system consisting in the first two equations.
- Complete the basis to a basis of the solution space of the linear system consisting of the first equation alone.
- Complete the basis to a basis of the total space .
We consider the last digit in the basis multiplication table as a family of -tuples of length , that is, the row vectors in the matrix
What is the dimensionMDLD/dimension (vs) of the linear subspaceMDLD/linear subspace in generated by these tuples?
===Exercise Exercise 8.17
change===
Let be a field,MDLD/field and let be a -vector space.MDLD/vector space Show that can not have a finite basisMDLD/basis (vs) and an infinite basis.
An -matrixMDLD/matrix over a fieldMDLD/field is called a magic square (or linear-magic square over ), when every column sum and every row sum in the matrix equals a certain number
.In this sense, the matrix
is, for every , a magic square.
Show that the set of all linear-magic squaresMDLD/linear-magic squares of length over a fieldMDLD/field is a linear subspaceMDLD/linear subspace in the space of all -matrices.MDLD/matrices
- Hand-in-exercises
Exercise (2 marks) Create referencenumber
Let be a field, and let be a -vector space. Let be a family of vectors in , and let
be the linear subspaceMDLD/linear subspace they span. Prove that the family is linearly independent if and only if the dimension of is exactly .
Exercise (4 (3+1) marks) Create referencenumber
a) Determine the dimensionMDLD/dimension (vs) of the solution space of the linear systemMDLD/linear system
in the variables .
b) What is the dimension of the solution space, if we consider the system in the variables ?
Exercise (4 marks) Create referencenumber
Show that the set of all real polynomialsMDLD/polynomials (field 1) of degreeMDLD/degree (polynomial) , 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.
Exercise (4 marks) Create referencenumber
Let be a field,MDLD/field and . Determine the dimensionMDLD/dimension (vs) of the space of all linear-magic squaresMDLD/linear-magic squares of length over .
Exercise (7 (3+2+1+1) marks) Create referencenumber
We consider the linear equationsMDLD/linear equations
over .
- Determine a basisMDLD/basis (vs) of the solution space of this linear system.
- Complete the basis to a basis of the solution space of the linear system consisting in the first two equations.
- Complete the basis to a basis of the solution space of the linear system consisting of the first equation alone.
- Complete the basis to a basis of the total space .
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