# Find the span of v_1,v_2,...,v_k,w when x_1 * v_1+x_2 * v_2+...+x_k *v_k=w has no solution

Find the span of ${v}_{1},{v}_{2},...,{v}_{k},w$ when ${x}_{1}\cdot {v}_{1}+{x}_{2}\cdot {v}_{2}+...+{x}_{k}\cdot {v}_{k}=w$ has no solution
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Ashly Sanford
Notice that
$\sum _{m=1}^{k}{x}_{m}{\mathbf{v}}_{m}=\mathbf{u}$
has only one solution. This would imply the set of vectors $\left\{{\mathbf{v}}_{1},\dots ,{\mathbf{v}}_{k}\right\}$ are linearly independent. Otherwise one can write one vector vi as linear combination of other vectors and there will be more than one solution (you may try to write out the other solutions explicitly to verify the above argument).
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Haven Kerr
Hint: Try to prove that if the vectors $\left\{{\mathbf{v}}_{1},\dots ,{\mathbf{v}}_{k}\right\}$ are linearly dependent then
$\sum _{m=1}^{k}{x}_{m}{\mathbf{v}}_{m}=\mathbf{u}$
has either no solution or more than one solutions for every $\mathbf{u}$ in $\mathbb{V}$. The above is just the contrapositive argument.