Show that a subset of a set of linearly independent vectors is linearly independent

set-theory linear-algebra vector-spaces

Students sometimes speak sloppily; often it doesn't matter much but here it seems to me the whole problem is caused by using the language imprecisely:

There is no such thing as an "independent vector"! There is such a thing as an independent set of vectors. If it were true that a set $S$ is independent if and only if every element of $S$ is an independent vector then it would be completely trivial that a subset of an independent set is independent. But that's not the definition...

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