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

linear combination problems and solutions in linear algebra

Problem 652

Prove that any set of vectors which contains the zero vector is linearly dependent.

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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.

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