linearly independent
a set of vectors v1, …, vp in R^n is linearly independent if the vector equation x1v1 + x2v2 + … xpvp = 0 has only the trivial solution x1=…= xp=0otherwise we say v1…vp is linearly dependent
In linear algebra, a set of vectors is said to be linearly independent if none of the vectors in the set can be expressed as a linear combination of the other vectors in the same set. In other words, the vectors are independent if no vector can be represented as a combination of the others, and each vector in the set adds some unique property to the set.
Symbolically, a set of vectors {v1, v2, …, vn} is said to be linearly independent if the only solution to the equation
a1v1 + a2v2 + … + anvn = 0
is a1 = a2 = … = an = 0, where a1, a2, …, an are scalars.
If there are scalars that satisfy the above equation, then the set of vectors is said to be linearly dependent. Linearly dependent vectors are ones where one of the vectors can be written as a linear combination of the other vectors in the set. This means that the set does not add any additional properties beyond what the individual vectors already provide.
Linear independence is an important concept in linear algebra and is foundational to many areas of mathematics, physics, and engineering. It is used extensively in solving systems of equations, finding bases for vector spaces, and in many other applications.
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