# OEF vector spaces --- Introduction ---

This module contains actually 16 exercises on vector spaces.

See also collections of exercises on definition of vector spaces or definition of subspaces.

### Two subsets

Let be a vector space. We have two subsets of , and , having respectively and elements. Answer:

• If , then .
• If , then .

### Two subsets II

Let be a vector space. We have two subsets of , and , having respectively and elements. Answer:

 If , is it true that  ? If , is it true that  ?

### Dim matrix antisym

What is the dimension of the (real) vector space composed of real antisymmetric matrices of size ×?

### Dim matrix sym

What is the dimension of the (real) vector space composed of symmetric real matrices of size ×?

### Dim matrix triang

What is the dimension of the (real) vector space composed of real triangular matrices of size ×?

### Dim poly with roots

What is the dimension of the vector space composed of real polynomials of degree at most , having as a root of multiplicity at least ?

### Parametrized vector

Let v1=() and v2=() be two vectors in . Find the value for the parameter t such that the vector v=() belongs to the subspace of generated by v1 and v2.

### Shelf of bookshop 3 authors

A bookshop ranges its shelf of novels.
• If one shows (resp. , ) copies of each title of author A (resp. author B, author C), there will be books on the shelf.
• If one shows (resp. , ) copies of each title of author A (resp. author B, author C), there will be books on the shelf.

How many titles are there in total for these three authors?

### Dim(ker) endomorphism

Let be a vector space of dimension , and an endomorphism. One knows that the image of is of dimension . What is the minimum of the dimension of the kernel of ?

### Dim subspace by system

Let E be a sub-vector space of R defined by a homogeneous linear system. This system is composed of equations, and the rank of the coefficient matrix of this system is equal to . What is the dimension of E?

### Generation and dependency

Let be a vector space of dimension , and let be a set of . Study the truth of the following statements.

 . . .

### Dim intersection of subspaces

Let be a vector space of dimension , and , two subspaces of with , . One supposes that and generate . What is the dimension of the intersection ?

### Image of vector 2D

Let be a linear map, with , . Compute , where . To give your reply, one writes .

### Image of vector 2D II

Let be a linear map, with , . Compute , where . To give your reply, one writes .

### Image of vector 3D

Let be a linear map, with , , . Compute , where . To give your reply, one writes .

### Image of vector 3D II

Let be a linear map, with , , . Compute , where . To give your reply, one writes .

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• Description: collection of exercises on vector spaces. serveur web interactif avec des cours en ligne, des exercices interactifs en sciences et langues pour l'enseigment primaire, secondaire et universitaire, des calculatrices et traceurs en ligne
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