Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 7



Exercise for the break

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




Exercises

Find, for the vectors

in , a non-trivial representation of the zero vector.


Find, for the vectors

in , a non-trivial representation of the zero vector.


Decide whether the following vectors are linearly independent.

  1. , , , in the -vector space .
  2. , in the -vector space .
  3. , in the -vector space .
  4. , in the -vector space .


Show that the three vectors

in are linearly independent.


Let be a -vector space, and let be a family of vectors in . Show that the family is linearly independent if and only if there exists a linear subspace such that the family is a basis of .


Determine a basis for the linear subspace


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

form 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


In , let the two linear subspaces

and

be given. Determine a basis for .


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 linear subspace. Show that has a basis consisting of vectors, such that all their entries are integer numbers.


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 , .


Let be a -vector space, let be a basis of , and let

be the corresponding bijective mapping in the sense of Remark 7.12 . Show that this mapping transforms the componentwise addition in into the vector addition in , that is,

holds.


Let be a basis of , and let

be the corresponding bijective mapping in the sense of Remark 7.12 . Show that this mapping is, in general, not compatible with componentwise multiplication in .


Let be a -vector space, and let , , be a basis of . Let , , be another family of vectors in . Suppose that, for every , the equality

holds. Show that also , , is a basis of .


Let be the polynomial ring over . For , set

Show that , , is a basis of .


Formulate and prove Theorem 7.11 for an arbitrary (not necessarily finite) family of vectors , .


We consider the real numbers as a -vector space. Show that the set of real numbers , where runs through the set of all prime numbers, is linearly independent. Tip: Use that every positive natural number has a unique representation as a product of prime numbers.


What does the Theorem of Hamel mean in the following examples?

  1. The real numbers as a -vector space.
  2. The set of all real seqeunces
  3. The set of all continuous functions from to .


Let be an ordered field, and let

be the vector space of all sequences in (with componentwise addition and scalar multiplication).

a) Show that (without using theorems about convergent sequences), the set of zero sequences, that is,

is a -linear subspace of .

b) Are the sequences

linearly independent in ?




Hand-in-exercises

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 (2 marks)

Show that, for the space of all -matrices , the matrices that have in position the entry , and elsewhere , 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 an -basis of .


Exercise (3 marks)

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


Exercise (4 marks)

Let be the polynomial ring over . We set , and, for , we set

Show that , , is a basis of .



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