#
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:

### Def dependence

Fill in:

### Def generation

Fill in:

### Dependence

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

### 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
.

### 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
,

which of the following subsets already generate
?
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- Description: elementary questions on vector spaces. interactive exercises, online calculators and plotters, mathematical recreation and games
- Keywords: interactive mathematics, interactive math, server side interactivity, algebra, linear_algebra, vector_space, basis, dimension, linear_systems