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

*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.

- For a family , , of elements in , linear span is a linear subspace of .
- 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.

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

- form a basis for .
- form a system of generators for .
- 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 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 .

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