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2011. december 12/14. Lineris algebra (A, B, C)
 

Summary: 2011. december 12/14.
Lineáris algebra (A, B, C)
13. el®adás
(vázlat)
K = C esetére átmenthetjük a 12/2 oldali tétel bizonyításának ötletét megnégyszerezve,
de 1./ felhasználása nélkül:
A(x + y, x + y) = A(x, x) + A(x, y) + A(y, x) + A(y, y),
A(x - y, x - y) = A(x, x) - A(x, y) - A(y, x) + A(y, y),
A(x + iy, x + iy) = A(x, x) - iA(x, y) + iA(y, x) + A(y, y),
A(x - iy, x - iy) = A(x, x) + iA(x, y) - iA(y, x) + A(y, y).
Ebb®l
A(x + y, x + y) = A(x, x) + A(x, y) + A(y, x) + A(y, y),
-A(x - y, x - y) = -A(x, x) + A(x, y) + A(y, x) - A(y, y),
iA(x + iy, x + iy) = iA(x, x) + A(x, y) - A(y, x) + iA(y, y),
-iA(x - iy, x - iy) = -iA(x, x) + A(x, y) - A(y, x) - iA(y, y).
Tehát
A(x + y, x + y) - A(x - y, x - y) + iA(x + iy, x + iy) - iA(x - iy, x - iy) = 4A(x, y),
végül
A(x, y) =
1

  

Source: Ágoston, István - Institute of Mathematics, Eötvös Loránd University

 

Collections: Mathematics