Exercise 6.8.4

Answers

1.

Yes. The form

f \times g

is bilinear and the integral operator is linear.

2.

No. If

J (x, y) \neq 0

for some

x

and

y

, then we have

H (cx, y) = {[J (cx, y)]}^{2} = c^{2} {[J (x, y)]}^{2} \neq cH (x, y) .

3.

No. We have

H (2, 1) = 4 \neq 2 H (1, 1) = 6 .

4.

Yes. The determinant function is an

n

-linear function and now

n

2

5.

Yes. When the field if

ℝ

, the inner product function is a bilinear form.

6.

No. It fails when

𝔽 = ℂ

. If we pick

V = ℂ

and choose the standard inner product. Thus we have

H (1, i) = ⟨ 1, i ⟩ = - 1 \neq iH (1, 1) = i .

jephianlin

2011-06-27 00:00