Familles libres

10:19 Feb 15, 2018
This question was closed without grading. Reason: Answer found elsewhere

French to English translations [PRO]
Mathematics & Statistics / Algèbre linéaire
French term or phrase: Familles libres
Algèbre 2
Etudier l'algèbre linéaire
Espaces vectoriels, sous-espaces, somme direct. ///////Familles libres,/////// familles génératrices, bases, théorème de la base incomplète. Dimension.

From document describing the courses for a Bachelor's degree "licence de sciences, technologies, santé - Mention mathématiques"

Thanks for your help with reference if possible.
Ghyslaine LE NAGARD
New Caledonia
Local time: 00:08

Summary of answers provided
5set of linearly independent vectors
Francois Boye
4linearly independent spanning set
Herbmione Granger

Discussion entries: 1



5 hrs   confidence: Answerer confidence 5/5
set of linearly independent vectors

Linearly independent vectors

Vector c is a linear combination of vectors a and b, because c = a + b. Therefore, the set of vectors a, b, and c is linearly dependent. Vectors d, e, and f are linearly independent, since no vector in the set can be derived as a scalar multiple or a linear combination of any other vectors in the set.

Note added at 1 day 20 hrs (2018-02-17 06:19:40 GMT)


Francois Boye
United States
Local time: 09:08
Specializes in field
Native speaker of: Native in FrenchFrench
PRO pts in category: 9

Peer comments on this answer (and responses from the answerer)
neutral  philgoddard: I have no idea whether this is correct, but your "explanation" is lifted from someone else's website.
22 mins
  -> This is the definition of linear algebra they teach all over the world
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2 days 23 hrs   confidence: Answerer confidence 4/5Answerer confidence 4/5
Famille libre
linearly independent spanning set

For your list, "linear independence" should work as well.

En algèbre linéaire, étant donnée une famille de vecteurs d'un même espace vectoriel, les vecteurs de la famille sont linéairement indépendants, ou forment une famille libre, si la seule combinaison linéaire de ces vecteurs qui soit égale au vecteur nul est celle dont tous les coefficients sont nuls.

This is a very typical use of the hypothesis that a set is linearly independent — obtain a relation of linear dependence and then conclude that the scalars must all be zero. The result of this theorem tells us that we can write any vector in a vector space as a linear combination of the vectors in a linearly independent spanning set, but only just. There is only enough raw material in the spanning set to write each vector one way as a linear combination. So in this sense, we could call a linearly independent spanning set a “minimal spanning set.” These sets are so important that we will give them a simpler name (“basis”) and explore their properties further in the next section.

Table of contents
1 Span, linear independence, basis and dimension
1.1 Span
1.1.1 Computing span in Rm
1.1.2 Minimal spanning subset
1.2 Linear independence
1.2.1 Determining linear independence in Rm
1.2.2 Maximal linearly independent subset
1.3 Basis and dimension
1.3.1 Definitions and basic results
1.3.2 Constructing bases from spanning sets and linearly independent sets
1.3.3 Dimension of a subspace

Note added at 3 days 7 hrs (2018-02-18 17:22:40 GMT)

-famille libre is linearly independent set
-famille génératrice is spanning set
-base is basis = linearly independent spanning set
You could use family, but we did not in the Linear Algebra course I took.

Herbmione Granger
Local time: 15:08
Native speaker of: Native in EnglishEnglish
PRO pts in category: 4

Peer comments on this answer (and responses from the answerer)
neutral  Francois Boye: You have confirmed the concept of 'linear independence of vectors'. So why do you reject?
2 days 5 hrs
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