Linear algebra help!!!!!!!!

Help me with this that T(x, y, z) = (2x-3y, 3y-2z, 2z) a linear transformation?

A transformation is linear if the following is true for some vectors u and v with constants a and b:

T(au + bv) = aT(u) + bT(v)

Let u = [i, j, k] and v = [l, m, n]. Then, working with the left side:

T(au + bv) = T([ai, aj, ak] + [bl, bm, bn])
T([ai, aj, ak] + [bl, bm, bn]) = T([ai + bl, aj + bm, ak + bn]}

Thus x = ai + bl, y = aj + bm and z = ak + bn, so sub those into the transformation definition:

(2(ai + bl) - 3(aj + bm), 3(aj + bm) - 2(ak + bn), 2(ak + bn))
(2ai + 2bl - 3aj - 3bm, 3aj + 3bm - 2ak - 2bn, 2ak + 2bn)

Now working with the right side of the transformation requirement:

aT(u) + bT(v) = aT([i, j, k]) + bT([l, m, n)]
a{2i - 3j, 3j - 2k, 2k} + b{2l - 3m, 3m - 2n, 2n}
{2ai - 3aj, 3aj - 2ak, 2ak} + {2bl - 3bm, 3bm - 2bn, 2bn}
{2ai - 3aj + 2bl - 3bm, 3aj - 2ak + 3bm - 2bn, 2ak + 2bn}

Thus LS = RS and the transformation is linear.

Done!