Exercise 6.3.1

Answers

1.

Yes. See Theorem 6.9.

2.

No. It just for the linear mapping

V \to 𝔽

. For example, the value of the identity mapping from

ℝ^{2}

ℝ^{2}

is not an element in

𝔽

3.

No. The equality holds only for the case

β

is an orthonormal basis. For example, let

A = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})

and

T = L_{A}

. Thus we have

T^{∗} = L_{A^{∗}}

. But for the basis

β = {(1, 1), (0, 1)},

we have

{[T]}_{β} = (\begin{matrix} 2 & 1 \\ - 1 & 0 \end{matrix}) \neq {[T^{∗}]}_{β} = (\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}) .

4.

Yes. See Theorem 6.9.

5.

No. Choose

a = i

b = 0

and

T = I_{ℂ} = U

. We have

{(aT)}^{∗} = \bar{a} T^{∗} \neq a T^{∗}

6.

Yes. See Theorem 6.10 and its Corollary.

7.

Yes. See Theorem 6.11.

jephianlin

2011-06-27 00:00