Question 2 1. Find the basis of the range and the kernel of the following linear trans- formations. (a) T: R3 R² such that T() = -1-2-3 -2-4-6 Z. [5] -x+y (b) T: R3 R³, such that T = 0 [4] -x+y 2 1-1 3 (c) T: RR3 such that T() = I. [5] 2 1 0 03 -6 2. Let PVR and Q: V→R be linear transformations, where V is a vector space. Define T: VR2 by T(v) = (P(v), Q(v)). (a) Show that T is a linear transformation. [4] (b) Show that ker T = ker Poker Q, the set of vectors in both ker P and ker Q. [5] 3. Let T (R2,S) →> (R2, 7) be defined by T x1-x2 Find the matrix M representing T when. (a) S= (b) S= = {(~)·()}• T = {(~)·()}· T= = {(²)·({}) } = {( -3)·(~})} (c) In each case (a) and (b) above, calculate T .T= by using M. [6] [6] ((²)) directly and [8]

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