9. Let T be an operator on an n-dimensional vector space V over a field F. (a) If T has n distinct eigenvalues prove that T is diagonalizable. If T is diagonaziable prove that the characteristic polynomial of T splits over F. (c) Give an example of an operator T such that the characteristic polynomial of T splits but T is not diagonalizable.

2. (a) Show that the list {p o(x), p 1(x), p 2(x),...,p n(x),...,} where Po(x) = 1, P₁(x) = 1 + x, pj(x) = 1 + x + · +x², is a basis of the vector space F[x] of all polynomials with coeficients in F. (b) What are the coordinates of the vector xn relative to the basis (po(x), p₁(x),..., pj(x), ...)?

at most 2. Find a basis {p(x), q(x)} for the vector space {f(x) = P₂[x] | f'(-8) = f(1)} where P₂[x] is the vector space of polynomials in x with degree You can enter polynomials using notation e.g., 5+3xx for 5 + 3x². p(x) = , 9(x) =

a) Consider a function space in (-∞, ∞) with orthonormal basis {n}. Show that any function f(x) expanded in this basis as f(x) = En Cnon(x), has expansion coefficients given by = (onlf). Cn = b) Now consider another function g(x) expanded in the above basis as g(x) = Σn dnon (1). Prove that (flg) = Σncdn. c) Now consider a non-orthonormal basis {x} for this space, such that (XmIXn) = Smn. Gen- eralise the previous result to find an expression for (flg) in terms of Snm. Confirm that if Smn = 6mn, then this result reduces to the above one.