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1. Let A be the matrix 40 0 6 1 3 -6 3 (a) By finding a basis of eigenvectors, determine matrices P and P-1 such that P-1AP is diagonal. (b) State the Cayley-Hamilton theorem, and verify it for the above matrix. 2. (a) Determine which of the following maps are linear (giving reasons for your answers). (i) f: R3-- R2 (X1,X2,X3) 7-- (x2 + 1,x3 - 1). (ii) f: R3-->R (x1,X2,X3) 7-- X1X2X3. (iii) f: R3--R3 (x1,X2, X3) 7-- (x1 + x2 + 2x3,0,x2). (b) Let {e1,ez,.,en} be the standard basis of Rnand f: R2 -- R3 be the linear map given on the standard basis by f(el) = 2ez + e3 andf(ez) = e1 + 3ez + 5e3. Determine the matrix of this map with respect to the bases {3e1 - ez,2e1+ 3ez} of R2 and {el, e1 + ez,e1 - ez - e3} of R3. (c) Is the map fin (b) as isomorphism? Give a reason for your