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 \(V\).
Sasa
Hakuna kifaa cha kupigia hesabu kinachoweza kufanya hesabu, lakini kifaa hicho kinaweza kutumiwa katika hesabu.
Keep your own working
Akaunti la bure laongezea mambo makuu juu ya kila somo, rekodi ya yale ambayo umemaliza, matatizo yako ya kutatua katika sehemu moja, na mtunzaji unayeweza kuuliza juu ya ukurasa huu.
Vaa alama yoyote ya ufafanuzi kamili, picha, na maana ya kila herufi.
Maana ya kila sehemu
Wazia hilo
Inastahili Kujua
Vielelezo
Pia, ufafanuzi huo ulitumiwa
Maswali ambayo watu huuliza
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.