linear independence
linear independence A fundamental concept in mathematics. Let x_{1}, x_{2},…, x_{n}
be mcomponent vectors. These vectors are linearly independent if for some scalars α_{1}, α_{2},…, α_{n},
implies α_{1} = α_{2} = … = α_{n} = 0
Otherwise the vectors are said to be linearly dependent, i.e. at least one of the vectors can be written as a linear combination of the others. The importance of a linearly independent set of vectors is that, providing there are enough of them, any arbitrary vector can be represented uniquely in terms of them.
A similar concept applies to functions f_{1}(x), f_{2}(x),…, f_{n}(x) defined on an interval [a,b], which are linearly independent if for some scalars α_{1}, α_{2},…, α_{n}, the condition, for all x in [a,b],
implies α_{1} = α_{2} = … = α_{n} = 0
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