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. - 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
.
- Description: collection of exercises on vector spaces. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games
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