Definition of a linear combination:

A vector v is a linear combination of vectors {v1, v2, ... , vk} if there are scalars c1, c2, ... , ck such that

v = c1v1 + c2v2 + ... + ckvk

Definition of linearly dependent:

A set of vectors {v1, v2, ... , vk} is linearly dependent if there are scalars c1, c2, ... ,ck with at least one not being zero such that

c1v1 + c2v2 + ... + ckvk = 0

In effect my question could also be viewed as: can the v from the definition of linear combinations be equal to the 0 vector?


The answer to the literal question in the title

Can a Linear Combination be Linearly dependent?

Is "no", because a "linear combination (of a list of vectors)" is a vector while the "linear independence (of a list of vectors)" is a property (so either true of false).

You can say that a list of vectors is linearly independent just when the only linear combination that's the $0$ vector is the obvious one with all $0$ coefficients.


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