let u and v be distinct vectors of a vector space V. Show that if {u,v} is a basis for V and a and b are nonzero scalrs, then both {u+v,au} and {au,bv} are also bases for V.
Please help, i dont even know where to start!!
well, i'll do the first one for you and you do the second one..
i) suppose and are linearly dependent. then there exists a nonzero scalar such that
then,
this means that which contradicts the fact that { } is a basis for .
hence, and must be linearly independent.
ii) Let . since { } is a basis for , we have for some scalars and in . In particular, let and where and are scalars in . Hence, . Thus, implies since we have written as a linear combination of . Clearly, . Therefore, .