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M, mklmk is orthogonal to every element of N. Consequently, it is orthogonal to the subspace generated by N. We leave the proof to the reader. 76) THEOREM(Gram-Schmidt). , m,} be a basis of the r-dimensional inner product space M. Then there exists an orthonormal basis {nl, . . , n,} such that the subspace generated by {ml, . . , mk} coincides with that generated by {n,, . . , nk} for 1 5 k < r. Proof. The proof is by induction on r. When M is one-dimensional, let n, be the vector ml/llml 11.

The words linear operator and linear mapping are frequently used synonyms for linear transformation. Let {ml, m 2 , m,} be a basis B of the three-dimensional vector space M. Let h be a mapping from M into K such that + cr2 m2 + u3 m,) = crl for each m whose expansion is ulml + cr2 m2 + u3 m3 in terms of the basis B. 3) Mm + m’) = h[(Ximi + a2m2 + u3m3) + ( B I ~ +I Pzmz + P3m3)I =~[(UI = cr, + + B&I + ( x 2 + BzImz + (u3 + P d m J = h(m) + h(m’). 4) h(cm) = ccr, = ch(m), c E K, m E M. Thus h is an element of Hom,(M, K ) .

C N, =M is a composition series for M. The fact that the dimension of N i , 0 _< i < r, is only one less than the dimension of N i + l implies that Ni is a maximal subspace of N i + l . 100) is a composition series of M. Since the dimension of N i , 0 < i 5 r, diminishes by one on passing from Ni to Ni-,, it follows that M has dimension r, in particular, M is finitely generated. The proof of the Jordan-Holder theorem is simple in the case of vector spaces where the theorem assumes the following form.

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2-transitive abstract ovals of odd order by Korchmaros G.

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