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Quizz vector spaces

Quizz vector spaces
--- Introduction ---

This module contains 11 training exercises on basic notions of
vector spaces.

Students may use them as a training tool for a better memorization of the lesson,
while teachers may put them into a work sheet with activated time limit, so that
proficiency can be tested.

Two subsets

Let be a vector space. We have two subsets of ,
and
, having respectively and elements. Answer:

If , then
.

If , then
.

Def dependence

Fill in:

Def generation

Fill in:

Dependence

Let
be a vector space,
a non-empty subset.

. True?

. True?

Dimension and elements

Answer:

Let
be a vector space,
a finite subset that . If ,
.

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?

Generated subspace

Let
be a vector subspace of
generated by a subset
of elements. What can be said of the dimension of
?

dim(
) is
equal to
.

Inclusion

Let
be a vector space,
,
two distinct finite subsets with
.

. True?

. True?

Set

Let
be a vector space of dimension ,
a subset of elements.

Can
be linearly independent?

Can
generate
?

Can
be a basis of
?

Set and basis

Let
be a vector space of dimension ,
a subset of
elements. Among the following properties, which ones ?

.

.

.

.

Generating subsets

Let
be a vector space generated by a set
= {
}. Given that there is a relation

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Description: elementary questions on vector spaces. Serveur Wims de l'ESPE-Nice-Toulon - Université de Nice - Sophia Antipolis

Keywords: interactive mathematics, interactive math, server side interactivity, algebra, linear_algebra, vector_space, basis, dimension, linear_systems