# 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 .

Other exercises on: vector spaces   linear algebra

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