Question

Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y,x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f Hence, or otherwise, show that: a vector subspace U-0 or U = V, if and only if, for every linear transformation V 4 v on V, U is an invariant subspace of f Hint: Extend a basis of U to a basis of V (c) Show that if λι, λ2 Ak are distinct eigenvalues of a matrix A and xi E Ελι[A] (i 1,2,..., n) then fxi,x2,... , xn) is a linearly independent set of vectors. Hint: Assume not and use the fact: given any set of finitely many dependent vectors there exists a smallest subset which is linearly dependent. (Why? Convince yourself with a separate proof of this statement.) (d) If v v be a linear transformation with dim Imf - k then f has at most + 1 distinct eigenvalues Hint: Use (c).

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