Linear Algebra I · Linear independence, bases, and dimension
Complete an independent family
Problem
Let $v:I\to V$ be linearly independent over a field $K$, and let $B$ be the basis obtained by adjoining new vectors to this family. Prove that $B$ is linearly independent and $\operatorname{span}_K(\operatorname{range}B)=V$.
Hint
Use the basis completion indexed by the original indices plus the new indices.
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