Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 7/en

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Warm-up-exercises

Exercise

Write in the vector

as a linear combination of the vectors


Exercise

Write in the vector

as a

linear combination of the vectors


Exercise

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 .


Exercise

Vektorraum/Erzeugendensystem und aufgespannter Unterraum/Fakt/Beweis/Aufgabe/en


Exercise

Show that the three vectors

in are linearly independent.


Exercise

Give an example of three vectors in such that each couple of them is linearly independent, but all three vectors are linearly dependent.


Exercise

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 und are linearly independent if and only if is not a scalar multiple of and vice versa.


Exercise

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 , .


Exercise

Determine a basis for the solution space of the linear equation


Exercise

Determine a basis for the solution space of the linear system of equations


Exercise

Prove that in the three vectors

are a basis.


Exercise

Establish if in the two vectors

form a basis.


Exercise

Let be a field. Find a linear system of equations in three variables, whose solution space is exactly




Hand-in-exercises

Exercise (3 points)

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 (2 points)

Establish if in the three vectors

form a basis.


Exercise (2 points)

Establish if in the two vectors

form a basis.


Exercise (4 points)

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 (3 points)

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




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