Exercise 6.8.3

Answers

1.
Check that
(H1 + H2)(ax1 + x2,y) = H1(ax1 + x2,y) + H2(ax1 + x2,y)
= aH1(x1,y) + H1(x2,y) + aH2(x1,y) + H2(x2,y)
= a(H1 + H2)(x1,y) + (H1 + H2)(x2,y)

and

(H1 + H2)(x,ay1 + y2) = H1(x,ay1 + y2) + H2(x,ay1 + y2)
= aH1(x,y1) + H1(x,y2) + aH2(x,y1) + H2(x,y2)
= a(H1 + H2)(x,y1) + (H1 + H2)(x,y2).
2.
Check that
cH(ax1 + x2,y) = caH(x1,y) + cH(x2,y)
= acH(x1,y) + cH(x2,y)

and

cH(x,ay1 + y2) = caH(x,y1) + cH(x,y2)
= acH(x,y1) + cH(x,y2).
3.
Pick H0(x,y) = 0 as the zero element and check the condition for a vector space.
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2011-06-27 00:00
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