linear independence
linear independence A fundamental concept in mathematics. Let x1, x2,…, xn
be m-component vectors. These vectors are linearly independent if for some scalars α1, α2,…, αn,
implies α1 = α2 = … = αn = 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 f1(x), f2(x),…, fn(x) defined on an interval [a,b], which are linearly independent if for some scalars α1, α2,…, αn, the condition, for all x in [a,b],
implies α1 = α2 = … = αn = 0
be m-component vectors. These vectors are linearly independent if for some scalars α1, α2,…, αn,
implies α1 = α2 = … = αn = 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 f1(x), f2(x),…, fn(x) defined on an interval [a,b], which are linearly independent if for some scalars α1, α2,…, αn, the condition, for all x in [a,b],
implies α1 = α2 = … = αn = 0
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linear independence