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Let U denote the vector space of continuous functions on the interval (0, xo), i.e. U={f:(0,xo) R I f is continuous}. For every a > 0 define the linear operator Ta : U U by Ta(f(x)) = f(ax). (a) Show that To o Tb = Tab- (b) Let W = span{1, x, In(x), . In(x) }. Show that Ta sends vectors in W into vectors in W and write the matrix [TalB of T in the basis B = {1, x, In(x), T In(x) 1. * (c) Find the inverse of the matrĂ­x [Ta]B from (b). I P6 Let A be a 7 X 7-matrix with characteristic polynomial EA (t) = - _t3 (t+3) (t-1) - (t-5) (t-6). (a) Show that the rank of A equals 4, 5, or 6. (b) Assume that the rank of A equals 4. Is A diagonalizable? Explain!

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