Question

7.3.1 Let U be a finite-dimensional vector space over a field F and T є L(U). Assume that λ0 E F is an eigenvalue of T and consider the eigenspace Eo N(T-/) associated with o. Let. uk] be a basis of Evo and extend it to obtain a basis of U, say B = {"l, . . . , uk, ul, . . . ,叨. Show that, using the matrix representation of T with respect to the basis B, the characteristic polynomial of T may be shown to take the form pra)- ) a, 7.3.38) where q(a) is a polynomial of degree I with coefficients in F. In particu- lar, use (7.3.38) to infer again, without relying on Theorem 7.6, that the geometric multiplicity does not exceed the algebraic multiplicity, of the eigenvalue Ao-

Answer 1

as a the basu where asi e matui k,n Cu kunJ Om-A ai, kai

以し kxl