# Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent ## Problem 652

Prove that any set of vectors which contains the zero vector is linearly dependent. Add to solve later

## Solution.

Let $\mathbf{0}$ be the zero vector, and $\mathbf{v}_1, \cdots, \mathbf{v}_k$ are the other vectors in the set.

Then we have the non-trivial linear combination
$1 \cdot \mathbf{0} + 0 \mathbf{v}_1 + 0 \mathbf{v}_2 + \cdots + 0 \mathbf{v}_k = \mathbf{0}.$ This is a non-trivial linear combination because one of the coefficients is non-zero.

Thus by definition, the set $\{ \mathbf{0} , \mathbf{v}_1 , \cdots , \mathbf{v}_k \}$ is linearly dependent. Add to solve later

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