so what are the implications of linear dependence+spanning the vector space? can you prove theorems or do problems more easily?

it seems powerful that if you have linear independence then each vector in the vector space can be written uniquely as a combination/summation of vectors? there's only one unique representation of each vector in the space?

for example S={i,j,i+j}, then S is linearily dependent since i+j+-(i+j)=0 and also (bi-ci)=0, but the subset T(S)=i, j is linearly independent. if S is linearly independent so is the subset T(S) this is equivalent to the statement if subset T(S) is linearly dependent so is S. so is dependence a bad quality for some reason?

then can you prove intuitively that a set of linearly independent vectors is a subset of basis?