Let u, v, and w be distinct vectors of a vector space V. Prove that if {u, v, w} is a basis for V, then {u + v, u + w, v + w} is also a basis for V. (Note: V is an arbitrary vector space, so you may not assume that u, v, and w are tuples.)

forms a basis for , which means any vector can be written as the linear combination,

Pick , , and ; then

Any can thus be written as a linear combination of the vectors , so these three vectors also form a basis for .

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