LetVandWbe vector spaces. Alinear transformationis a functionT:VWsuch that for all u,vVand all cR:

T(u+v) =T(u) +T(v) andT(cv) =cT(v)

Let B= {b1,b2, ...,bn}be a basis for a vector space V. Let C: VR^nbe theB-coordinate mappingC(v) = [v]B. Prove thatCis a linear transformation.

Hint:SinceBis a basis forV, every vectorvVcan be written uniquely as a linear combination of the vectors inB:

v=c1b1+c2b2++cnbn