#
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. 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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