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:

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.

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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