85年政大應數碩班考線性代數

Let C be the space of all real-value continuous functions defined on [-pi,
pi]. For f and g in C. define

(f,g) = 積分 pi 到 -pi f(x)g(x)dx

(1) show that (,) is an inner product on C.
(2) show that {1,sinx,cosx,sin2x,cos2x} are mutually orthogonal.
(3) find a finction in the span of {1,sinx,cosx,sin2x,cos2x} that is closest
to the function g(x) = |x| on [-pi, pi].
(4) sketch the graphs of g and the function that you found in (3) on the
same coordinate system;

5,5,10,5

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