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 v_{1}=() and v_{2}=() be two vectors in
. Find the value for the parameter t such that the vector v=() belongs to the subspace of
generated by v_{1} and v_{2}.

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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