Let be a basis of . Consider the family of vectors  defined by

,

,

,

.

Find the dimension of .

Let be the set of all solutions to a homogeneous linear system of five variables and three equations. Then,

Let be a finite dimensional vector space, a basis of and a family of vectors of . Denote by the matrix associated to in the basis . Then, is a spanning set if and only if

Let be the vector space of functions defined over . Denote by the set of all functions that vanish at least once and by the set of all functions that are bounded from above.

("vanish"= "s'annuler" ; "bounded from above"="majorée")

Consider the polynomials , , . We admit that the family is a basis of . Then, the coordinates of in the basis are:

Let be a finite dimensional vector space and let and be two families of vectors of such that and . Among the following propositions, which one is true?

Find the dimension of the space of solutions to the following linear system:

Let be a basis of . Consider the family of vectors  defined by

,

,

,

.

Find the dimension of .

Let be the set of all solutions to a homogeneous linear system of six equations and four variables. Then,

Let be a finite dimensional vector space, a basis of and a set of vectors of . Denote by the matrix associated to in the basis . Then, is a linearly independent set if and only if