The Linear Model and Hypothesis by George Seber

Author:George Seber
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

and find what weights we must use for H 1, H 2, and H 3 to be orthogonal with respect to .

Theorem 6.7

The hypotheses H i (i = 1,2,3) are orthogonal with respect to G if and only if and for all i and j.

Proof

We shall use a more direct proof than that in Theorem 6.5. (a) Necessity. From the alternative form of the definition of orthogonality we have that the hypotheses are orthogonal if and only if for j ≠ i. Thus a necessary condition for orthogonality is that . The vector space is defined by the set of such that θ ⊥ ω 3, i.e.,

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