Show that any subset of a linearly independent set is also linearly independent.

Question

anonymous · Answer

I would do this by proof of contradiction.  Assume that you have a linearly independent set of vectors:
$$\left\{ b_1, b_2,b_3,\ldots ,b_n \right\}$$

and some subset of them:
$$\left\{ b_1, b_2,\ldots, b_k \right\}$$
is linearly dependent.   How does that mess things up?

anonymous · Answer

Is it because then the coefficients of the linearly dependent vectors will equal to a non-zero constant?

anonymous · Answer

Thats right.  If we had a subset of those vectors who are dependent, then there would be some linear combination of them that equals zero, where the coefficients in front of them are not all zero.  

If that were true, then the same linear combination would make the supposedly linearly independent set dependent instead.

anonymous · Answer

So thats the proof lol.  Every subset of a set of linearly independent vectors must also be linearly independent, because if there existed a subset that wasnt linearly independent, the  main set would also not be linearly independent.

anonymous · Answer

oh i see. all right. i understand now. thanks (: