Find description of rt as the intersection of hyperplanes

Question

(1) Consider the linear transformation T : R⁴  -> R³ dened by T(x, y, z, w) = (x + 2y   z + w;  x   2y + z   2w; x + 2y   z):

(a) By directly using the denition of the range of a linear transformation, write down a description of the range R(T); and determine a nonzero vector in it.

(b) Find description of R(T) as the intersection of hyperplanes and deduce a basis for R(T) and the rank r(T):

(c) Find a basis for ker(T) and determine n(T) the nullity of T:

(d) Verify the the theorem T : V  ! W, then dim(V ) = r(T)+n(T) for the linear transformation considered above