The set T = {u + v + w, u + 2v + 3w, 2u + 3v + 4w, u - 4v + 2w } is LD,
because 2u+3v+4w = (u+v+w)+(U+2v+3w).
Suppose that u, v, w are vectors in such that
S = {au + bv + cw, du + ev + fw, gu + hv +iw} is linearly independent for some numbers a,b,c,d,e,f,g,h, and i. Prove that u,v,w are linearly independent.
Is the set T = {u + v + w, u + 2v + 3w, 2u + 3v + 4w, u - 4v + 2w } linearly dependent or linearly independent?
In that case I would put my vectors into a matrix. Then I will try to determine whether the homogeneous linear system has non-trivial solutions. Keep in mind that a homogeneous linear system has only the trivial solution iff its column vectors are linearly independent. There are ways to know whether the homogeneous system has the trivial solutions.
You can show that the matrix is invertible, therefore the homogeneous linear system has ONLY the trivial solution.