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Linear independence
Introduction
To look more closely at the structure of vector spaces,
we defined spanning set of a vector space.
In particular, it is desirable to find a minimal spanning
set set with no unnecessary elements.with no unnecessary elements.
Introduction
To find a minimal spanning set, we need to konw how the
vectors in the collection depend on each other.
In this section we introduce the concepts of linear
dependence and linear independence.dependence and linear independence.
These simple concepts provide the keys to understanding
the structure of vector spaces.
We will show some examples and discuss the properties of
linearly independent vectors.
Outline
1. Minimal spanning set
2.2. Definition of linear dependence and linear Definition of linear dependence and linear
independence
3. Examples
4. Properties of linearly independent vectors
Minimal spanning set
A minimal spanning set is a spanning set with no
unnecessary elements, i. e., all the elements in the set
are needed in order to span the vector space.
It contains the smallest possible number of vectors.
TT 33
{e{e , , ee , , ee , (1, 2, (1, 2, , 3)3) } is not a minimal spanning set of R} is not a minimal spanning set of R ..
1 2 3
To see how to find a minimal spanning set, it is
necessary to consider how the vectors in the collection
depend on each other. Consequently, we introduce the
concepts of linear dependence and linear independence.
Two observations:
(I) If v , v , ···, v span a vector space V and one of
1 2 n
these vectors can be written as a linear combination of
the other n − 1 vectors, say v , not needed to span V
i
then those n − 1 vectors span V.
(II) Given n vectors v , v , ···, v , it is possible to write
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