Linear independence
A set of vectors in a vector space is linearly
dependent if one of them can be written as a linear combination
of the others – if it "depends" on the others. Vectors
which are not linearly dependent are linearly
independent.
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: look at three different
vectors in three-dimensional Euclidean space. (The gray vectors are just i,
j and k for
reference.)
If any two of the vectors are
parallel, then one is a scalar multiple of the other. A scalar multiple
is a linear combination, so the vectors are linearly
dependent.
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If no two of the vectors are parallel but all three lie in a plane, then any
two of those vectors span that plane. The third vector is a linear combination
of the first two, since it also lies in this plane, so the vectors are
linearly dependent.
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If the three vectors don't all lie in some
plane through the origin, none is in the span of the other two, so none
is a linear combination of the other two. The three vectors are linearly
independent.
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If you have a large collection of vectors, checking for linear independence
by checking separately whether each vector is a linear combination of the others
could be very tedious. Fortunately, there's a way to do all the checks at once.
The vectors v1,
v2,
..., vk
are linearly independent if and only if the linear
dependence equation
c1v1
+ c2v2
+ ... + ckvk
= 0
has only the trivial solution
c1
= c2 = ... = ck
= 0.
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The linear dependence equation always has at least the trivial solution
c1 = c2
= ... = ck = 0. We're only interested
in non-trivial solutions.
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Suppose it also has a solution with one of
the ci's not zero, say c1
≠ 0. Then you can solve for v1
as a linear combination of the others:
v1
= –(c2/c1)v2
– (c3/c1)v3
– ... – (ck/c1)vk,
so the vectors are linearly dependent.
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On the other hand, suppose the vectors are
linearly dependent, with (say) v1
a linear combination of the others:
v1
= c2v2
+ ... + ckvk.
Then (–1)v1
+ c2v2
+ ... + ckvk
= 0, so the linear dependence equation
has a solution with some ci not zero
(specifically c1 = –1). |
| To determine whether or not a set of vectors
is linearly independent |
Set up the linear dependence equation for those vectors and solve it.
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If you get only the trivial solution
(all coefficients zero), the vectors are linearly independent.
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If you get any solution other than
the trivial solution, the vectors are linearly dependent.
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. (You can check
these by setting up a linear dependence relation and solving for the coefficients.)
The elementary vectors in Rn
are linearly independent.
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The matrices
in the space of 2 x 2 matrices are linearly independent.
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The polynomials 1, x, x2,
x3, ..., xn
are linearly independent.
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