Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 8
- Exercise for the break
Matrices/2x2/Dimension/Exercise
- Exercises
Linear system/Dimension/1/Exercise
Space of matrices/Dimension/Exercise
Diagonal matrices/Linear subspace and dimension/Exercise
Symmetric matrices/Linear subspace/Dimension/Exercise
Upper triangular matrices/Linear subspace and dimension/Exercise
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.
Vector space/Finite dimension/Linear subspace of full dimension/Exercise
Parametrized vectors/abc/Cyclically swapped/Dimensions/Exercise
Let be a field, and let and be two finite-dimensional vector spaces with
and
What is the dimension of the Cartesian product ?
Linear subspaces/Sum of the dimensions larger/Intersection/Exercise
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 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 change theorem. Can one take any standard vector?
Linear system/Basis completion/1/Exercise
Basic multiplication table/Last digit/Vector space dimension/Exercise
Infinite basis/Finite basis/Exercise
Magical square/Linear/Definition
In this sense, the matrix
is, for every , a magical square.
Magical squares/Linear/linear subspace/Exercise
- Hand-in-exercises
Exercise (2 marks)
Let be a field, and let be a -vector space. Let be a family of vectors in , and let
be the linear subspace they span. Prove that the family is linearly independent if and only if the dimension of is exactly .
Linear system/Dimension/2/Exercise
Exercise (4 marks)
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.
Magical squares/Linear/Dimension/Exercise
Linear system/Basis completion/2/Exercise
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