Building bases
Knowing the dimension of the vector space you're dealing with can be very
useful when you're trying to find bases or decide if a given set is a basis.
Suppose you have a collection
of vectors in an n-dimensional
vector space. Then |
if you have more than n vectors,
they must be linearly dependent
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if you have fewer than n vectors,
they cannot span the space
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if you have exactly n vectors,
they may or may not form a basis, but you need
check only one of the conditions (linear independence or spanning)
to check for a basis – the other condition then follows automatically.
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.
(omitted)
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You can also "thin out" a spanning set or "top up" a linearly
independent set to get a basis.
For a given set of vectors in a finite-dimensional
vector space V: |
if the vectors span V but are
not linearly independent, you can always remove dependent vectors from
the set to get a basis of V
|
if the vectors are linearly independent but don't span V,
you can always include other vectors outside their span to get a basis
of V.
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.
(omitted)
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How you actually do this is different for different spaces. In Rn,
you can use the dependency algorithm
| To reduce a spanning set in Rn
to a basis |
Use the dependency algorithm to find those vectors in the set which
are linearly dependent on the others and remove them.
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| To extend a linearly independent set in Rn
to a basis |
Include the elementary vectors in your set. The enlarged set of vectors
is now a spanning set for Rn.
|
Use the dependency algorithm to remove the dependent vectors from this
spanning set to get a basis. (Make sure you put the elementary vectors
on the right, so the algorithm chooses your original linearly independent
vectors first.)
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