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Suppose that there is some vector. And the vector can be decomposed into linear combination of linearly independent vectors. If we set one of coefficients of these vectors to be zero, and we get the vector, would there be another linear combination with all coefficients non-zero that describes the vector?

Also, are all linear combinations usually set to have non-zero coefficients?

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To answer your first question, no. A linear combination of linearly independent vectors uniquely describes a given vector. This is because two vectors are equal iff their components (i.e. the coefficients on whatever unit vectors you are using) are equal.
For your second question, the answer is also no. There is nothing wrong with having some coefficients as zero. For instance, representing the unit vectors (whatever you define them as) in your system requires each of the other coefficients to be zero.

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