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(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
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