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.

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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