This is a problem from Steven Levandosky\'s \"Linear Algebra\"textbook: \"Let {u

Question

This is a problem from Steven Levandosky's "Linear Algebra"textbook:

"Let {u,v,w} be a linearly independent set. Show that {u+v,u+w, v+w} is a linearly dependent set."

My approach to the problem was to establish what I do know and seeif that would spark some ideas for a proof or explanation of why{u+v, u+w, v+w} is a linearly dependent set. I know:

-u, v, and w are not collinear
-none of the vectors made by u+v, u+w, v+w will be u, v, or wbecause it is given that the original set is linearlyindependent
-vectors u, v, and w must be in R^3. If they were in R^2,then they could not be linearly independent.

I still haven't quite figured out why this means that {u+v, u+w,v+w} must be linearly independent.

Explanation / Answer

I solved it on my own :-) The trick is to write out theequation as if the problem were linearly independent and to thenrewrite it and obtain new constants next to the old vector set.

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