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If there is vectors, $ v_1,v_2,...$ .

They are linearly independent if $c_1v_1+c_2v_2+... = 0$ with $c_1=c_2=...=0$.

  • if $v_1,v_2,...$ are linerly independent and so what?

  • what will the linear independence of vectors lead to? Base?

  • Do linear independence has any connection to dot product or orthogonality?

  • what will happen if they are not linearly independent?

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Linearly independence means $c_i=0$ is the only way to achieve $c_iv_i=0$. It follows you can't write any one vector as a linear combination of the others, so the space they span together can't be spanned by any proper subset of them. In an inner product space, you can make an orthogonal basis from independent vectors, but they're not guaranteed to be one.

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Linear independence means that no vector in the set can be deduced from the others, so if you drop one, the span (i.e. the space that can be built on these vectors) reduces.

E.g. in 3 space, if three vectors are linearly dependent, they are coplanar. If you drop one, the span is still a plane.

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  • $\begingroup$ Even in a linearly dependent set it may be possible to drop a vector and have the span be reduced. Although if this is true for every vector, then the set is linearly independent. $\endgroup$ Oct 29 '19 at 11:08
  • $\begingroup$ @MeesdeVries: you are right, but I didn't want to clutter my answer with details. $\endgroup$
    – user65203
    Oct 29 '19 at 11:10

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