Absorption and Space Reddening in the Galaxy as Shown by the by Stebbins J. PDF

By Stebbins J.

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V , ) are z , . Therefore A v = Av. 2 (Spectral Theorem). If A E C(V, V) is self-adjoint, then there exists an orthonormal basis ( x , ,. . , x , ) for V and real numbers A,, . . , A, such that Further, A , , . . , A , are the eigenvalues of A and Ax, = A,x,, i = 1,. . , n. Proof: The proof of the first assertion is by induction on dimension. For n = 1, the result is obvious. Assume the result is true for integers 1,2,. . , n - 1 and consider A E C(V, V), which is self-adjoint on the inner product space (V, (.

For (V, ( . , Ax,) is an orthonormal basis. Now, assume (iv) holds. For x E V, we have = C(x, xi)'. Thus x = C(x, xi)x, and (1x11~ ,))a llAx1I2 = (Ax, Ax) = (x, x,)Ax,, i ( x , x,) Ax, Therefore (ii) holds. Some immediate consequences of the preceding proposition are: if A is orthogonal, so is A-' = A' and if A , and A, are orthogonal, then AlA2 is orthogonal. Let B(V) denote all the orthogonal transformations on the inner product space (V, ( -, . )). Then Q(V)is closed under inverses, I E B(V), and O(V) is closed under products of linear transformations.

Follows. That {win x,li = 1,. . , n, j = 1,. . , m) is an orthonormal basis for (C(V, W), ( . )) follows immediately from the definition of ( . , a) a ) . A few words are in order concerning the inner product ( . ) on C(V, W). Since {win x,li = 1,. . , n, j = 1,. . , m) is an orthonormal basis, 32 VECTOR SPACE THEORY we know that if A E C(V, W), then since this is the unique expansion of a vector in any orthonormal basis. 24. Thus (A, win x,) = [w,, Ax,] for i = 1,. . , n and j = 1,.

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Absorption and Space Reddening in the Galaxy as Shown by the Colors of Globular Clusters by Stebbins J.


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