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 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 subvector 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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