Careful : in this question, checking wrong answers will yield negative points!

Let and be two finite dimensional vector spaces, a linear map, and two bases of and respectively. You are given

.

Among the following claims, choose all the ones that are correct.

Compute the following determinant

Let be a square matrix with 3 rows and 3 columns. Let . Knowing that , what is the value of ?

Let and be two vector spaces, and two linear maps. Moreover, suppose that . Among the following claims, which one is the correct one? (only one possible answer)

Consider the linear map defined by .

Give the dimension of .

Careful : in this question, checking wrong answers will yield negative points!

Consider the linear map whose associated matrix in the canonical bases is

Among the claims below, choose all the correct ones.

Let be a vector space and two families of vectors of . Denote by and and consider the following propositions:

: " is a basis of ''

: " and are supplementary in ".

Then, we have

Let be the vector space of all sequences. Denote by the set of all arithmetic sequences and by the set of all geometric sequences. Then

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

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