## Related questions with answers

Question

Suppose that $\mathcal{V}=\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{m}\right\}$ spans a vector space V, and suppose that v is in V but not in $\mathcal{V}$. Prove that $\left\{\mathbf{v}, \mathbf{v}_{1}, \dots, \mathbf{v}_{m}\right\}$ is linearly dependent.

Solution

VerifiedSince $v\notin \mathcal V$, but $v\in V$ and the fact that $\mathcal V$ spans $V$ we can conclude that $v$ must be a linear combination of vectors from $\mathcal V$.

Thus, by theorem 7.8 we know that $\left\{ v,v_1,\dots,v_m\right\}$ is linearly dependent.

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