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Advanced calculus by Spencer, Donald Clayton; Nickerson, Helen Kelsall PDF

By Spencer, Donald Clayton; Nickerson, Helen Kelsall

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If PP = P. An endomorphism Show that P : V -> V is called a projection ker P n im P = TIV. Show that each vector A of V is uniquely representable as the sum of a vector of 48 ker P and a vector of im P. ker Q = im P, 1m Q projection, and that Q r I - P is also a Show that = ker P. Show that ~· PQ = QP Let 4. dim V be finite. Show that any endomorphism T of V can be expressed as a composition SP where projection and 1f JJ S is an automorphism. An endomorphism 5. = I. Show that, 1f is an involution.

Ah) A1, •.. • , Ah· ~· vectors Let U be the linear subspace spanned by the Ai - A1 for i = 2, 3, ••• , h. • , . Ah) = A1 + U (3 ) A vector on the left has the form (2). We assert that zxiAi where the h But (2) implies x 1 = 1 - zi= 2 xi, and thus is a vector of A1 + u. x•s satisfy so Conversely a vector of A1 + U has the form The stnn of the coefficients on the right is is in E(A 1, ... ,Ah). 1; hence the vector This proves (3), and the first conclusion of the proposition. Since have Ai = Al + (Ai - A1 ), and Ai - A1 is in Ai e A1 + u, so Ai e E(A 1, •.

The matrix representation M : L(V, W) -> Rkn is an isomorphism. Proof. If T is given by (3) above, and x £ R, 42 (xT)(Ai) = xT(Ai) = ncajiBj' then xM(T). M(S) = (~ji)' If Therefore M(xT) then S(Ai) + T(Ai) = I:ajiBj + E~jiBj E(aji + ~ji)Bj · Therefore M(S + T) = M(S) M(T). + We have seen already in §5 that This proves that M is linear. M is bijective, and hence is an isomorphism. 3. let S Proposition. Let U ~-> V be linear. The induced function S* : L(V, W) defined by S* (T) = TS Similarly, if T : V ~-> T*(S) = TS -> L(U, W) T L(V, W), for each E is linear.

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Advanced calculus by Spencer, Donald Clayton; Nickerson, Helen Kelsall

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