a Either find vectors u v w in R3 such that uvw v w2vu span

a) Either find vectors u, v, w in R3 such that {u+v-w, v, w+2v-u} span R3 or justify their non-existence.

b) Suppose T: R2 -> R2 is an R-linear transformation. Do there exist nonzero vectors u, v, w in R2 such that T(u) + u = T(v) + 2v =T(w) + 3w = 0?

Solution

a.

No such vectors exist because

-(u+v-w)+3v=w+2v-u

HEnce vectors are linearly dependent. And for a set of vectors to span R3 we need a minimum of 3 linearly independent vectors.

Hence proved

b.

T(u)=-u

T(v)=-2v

T(w)=-3w

No there does not exist such vectors because T is transformation from R2 to R2 hence represented by a 2x2 matrix so can have at most 2 eigenvalues

But if such u,v,w exist then it will have three eigenvalues,-1,-2,-3 which is not possible.