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Say, $S={v_1, v_2, ... , v_k}$ where $v_i$ is vector ($i$ is from $1$ to $k$)

and if some $v_j$(say, $v_2, v_3, v_4, v_5$) are zero vectors and every other vectors are linearly independent each other, then is $S$ linearly independent or dependent?

In other words, does some vectors which are zero vector affect the linear independence in a set, like $S$?

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A zero vector can be considered linearly dependent on any vector as $v \cdot 0 = 0$ for any vector $v$.

Thus your set $S$ would be linearly dependent, with dimension $k-4$ (since you said all the non-zero vectors were linearly independent).

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