Let S = \{v_1, v_2, \ldots, v_n\} be a set of vectors in a vector space V.
Linear Independence
Let \(S = \{v_1, v_2, \ldots,
v_n\}\) be a set of vectors in a vector space \(V\).
If there exist scalars \(\alpha_1, \alpha_2 \ldots \alpha_n \in F\) such that not all of the \(\alpha_i\)'s are zero and
\[\begin{aligned}\end{aligned}\],
then \(S\) is said to be linearly dependent.
If the set \(S\) is not linearly dependent,
then it is said to be linearly independent.
More specifically, \(S\) is a linearly independent set if
\[\begin{aligned}\end{aligned}\]
implies that
\[\begin{aligned}\end{aligned}\]
for any set of scalars \(\{ \alpha_1, \alpha_2 \ldots \alpha_n \}\).
The definition of linear dependence makes more sense if we consider the following proposition.
The following proposition is a consequence of the fact that any system of homogeneous linear equations with more unknowns than equations will have a nontrivial solution.
We leave the details of the proof for the end-of-chapter exercises.
A set \(\{ e_1, e_2, \ldots,
e_n \}\) of vectors in a vector space \(V\) is called a
basis for \(V\) if
\(\{ e_1, e_2, \ldots,
e_n \}\) is a linearly independent set that spans \(V\).
Indi sen
Hiç bir hasapçy buny çözmez, emma onuň bölekleri hasaplanmak üçindir. Aşakda birini synla ýa-da özüňi ýaz.
Özüň işle
Mugt hasap her bir ders üçin notlar goşýar, tamamlanan zadyňyzyň ýazgysyny, çözen problemaňyzy bir ýerde, we bu sahypa barada sorap biljek mugallymy. Matematika özüçe her kim üçin açyk, girseň ýa girmeseň.
Eň täze düşündirme, surat, we her harpyň manysy üçin her bir simwoly basyň.
Her bir bölegiň manysy
Resim
Bilen
Mysal
Bu beýanda hem ulanylan
Adamlar soraýan soraglar
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.