Linearly Independent Vectors

In linear algebra, a set of vectors in a vector space is said to be linearly independent if none of the vectors in the set can be expressed as a linear combination of the others.

More formally, given a vector space V over a field F, a set of vectors {v1, v2, ..., vn} in V is linearly independent if and only if the only scalars c1, c2, ..., cn in F that satisfy the equation:

c1v1 + c2v2 + ... + cnvn = 0

are the trivial scalars c1 = c2 = ... = cn = 0.

In other words, the only way to obtain the zero vector as a linear combination of the vectors in the set is by taking all the coefficients to be zero.

Geometrically, linearly independent vectors can be thought of as vectors that point in different directions, meaning that no vector in the set can be obtained by scaling or adding a combination of the others.

The concept of linear independence is fundamental in linear algebra and has many important applications, such as determining the dimension of a vector space, finding bases of vector spaces, and solving systems of linear equations.

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#linearlyindependent
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