Thanks, i also have a question other than part (b) above,
say if u,v,w is lin,indep
q=w-v+lu and l E R
show u,v,q is lin,indep.
I can see u,v,q is lin,indep if q=w-v+u
But what if the equaition at above with a l real number multiply by u in the equation?
No, if there exist any numbers, a, b, c, not all 0, so that au+ bv+ cw= 0 then they are dependent if u+ v+ w= 0 then a= b= c= 1 are not all 0 so they are dependent. Of course, with the u, v, w you give here, u+ v+ w is NOT 0.
If 3 vectors are independent, they they form a basis for a 3 dimensional space.And how do i find out if its form a basis of R^3
Suppose u, v, q were NOT linearly independent. The there exist a, b, c, not all 0, such that au+ bv+ cq= 0. But q= w- v+ lu so that is au+ bv+ c(w- v+ lu)= (a+ lc)u+ (b- c)v+ cw= 0. Since u, v, and w are independent, we must have a+ lc= 0, b- c= 0, and c= 0. Those reduce to a= 0, b= 0, and c= 0 contradicting the assertion that they are not all 0. Since c= 0, the "l" in lc does not matter.