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 \(V\).
Şimdi sen
Hiçbir hesap makinesi bu soruyu çözemez, ama parçaları hesaplanabilir. Aşağıdakilerden birini deneyin veya kendinizinkini yazın.
Kendi çalışmalarınızı saklayın
Ücretsiz bir hesap her ders için notlar, bitirdiğiniz şeylerin kaydı, çözüldüğünüz problemlerin bir yerdeki kaydı ve bu sayfa hakkında soru sormak için bir öğretmen ekler. Matematik kendisi, giriş yapmış veya yapmamış herkese açıktır.
Tüm tanım, resim ve her harfin anlamı için herhangi bir sembolü tıklayın.
Her parçanın anlamı nedir?
Düşünsene.
Bilmeye değer.
Örnekler
Bu açıklamada da kullanılıyor.
Sık sorulan sorular
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.