Let S u, v be a linearly independent set. Prove that the {u + v,u - v} is linearly independent

Answer with explanation:It is given that {u,v} be a linearly independent set of a set S.This means that there exist constant a,b such that if:                                 au+bv=0                              then a=b=0Now we are asked to prove that:{u+v,u-v} is a linearly independent set.Let us consider there exists constant c,d such that:                             c(u+v)+d(u-v)=0To show:   c=d=0The expression could also be written as:  cu+cv+du-dv=0( Since, using the distributive property)Now on combining the like terms that is the terms with same vectors.cu+du+cv-dv=0i.e.(c+d)u+(c-d)v=0Since, we are given that u and v are linearly independent vectors this means that:c+d=0------------(1)and c-d=0 i.e c=d-----------(2)and from equation (1) using equation (2) we have:2c=0i.e. c=0and similarly by equation (2) we have:          d=0Hence, we are proved with the result.We get that the vectors {u+v,u-v} is linearly independent.