Jump to content

Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 23

From Wikiversity



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.

  1. For a family , , of elements in , linear span is a linear subspace of .
  2. The family , , is a spanning system of if and only if


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

and

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.

  1. If the family is linearly independent, then for each subset , also the family  , is linearly independent.
  2. The empty family is linearly independent.
  3. If the family contains the null vector, then it is not linearly independent.
  4. If a vector appears several times in the family, then the family is not linearly independent.
  5. A vector is linearly independent if and only if .
  6. 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


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

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.

  1. form a basis for .
  2. form a system of generators for .
  3. 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 linear 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

and

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

Exercise (3 marks)

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.


Exercise (4 marks)

We consider in the linear subspaces

and

Show that .


Exercise (2 marks)

Establish if in the three vectors

form a basis.


Exercise (2 marks)

Establish if in the two vectors

form a basis.


Exercise (4 marks)

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 .


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.


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 .



<< | Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I | >>
PDF-version of this exercise sheet
Lecture for this exercise sheet (PDF)