OEF vector subspaces
--- Introduction ---
This module gathers for the time being 8 exercises on
subspaces of vector spaces.
Dimension of intersection
Fill-in: Let F be a vector space of dimension , and let
,
be two vector subspaces of F, of dimensions respectively and . Then
is at least
and at most
.
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?
Dimension of sum
Fill-in: Let F be a vector space of dimension , and let
,
be two vector subspaces of F, of dimensions respectively and . Then
is at least
and at most
.
Subbase
Fill-in: let F be a vector space of dimension , and let B be a basis of F. Let
be a subset of de elements, and let E be the vector subspace of F generated by
. Then dim(E) is
equal to
.
Subbase II
Fill-in: let F be a vector space of dimension , and let B be a basis of F. Let
and
be two subsets of B, with respectively and elements. Suppose that
has elements. Let
and
be the vector subspaces of F generated respectively by
and
, and let
.
Then
is
equal to
.
Dimension of subspace
Fill-in: Let E be a vector subspace of
. Then dim(E) is
equal to
.
Dim subspace of matrices
Fill-in: let M× be the vector space over
of × matrices, and let E be the vector subspace of M× consisting of matrices A such that =0, where B is a fixed non-zero matrix of dimension ×. Then dim(E) is at least
, and at most
.
Extension of subspace
Let F be a vector space of dimension , E a subspace of F generated by a set S, with
. Let v be a vector of F which a linear combination of vectors in S, and let
be the vector subspace of F generated by
. What is the dimension of
?
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- Description: collection of exercises on vector subspaces. serveur web interactif avec des cours en ligne, des exercices interactifs en sciences et langues pour l'enseigment primaire, secondaire et universitaire, des calculatrices et traceurs en ligne
- Keywords: math, interactif, exercice,qcm,cours,classe,biologie,chimie,langue,interactive mathematics, interactive math, server side interactivity,exercise,qcm,class, algebra, linear algebra, vector space, subspace, base, dimension, linear system