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Linear Algebra

Exercise 6.3.16

Answers

1.

Compute

{⟨ x, {(T + U)}^{∗} (y) ⟩}_{1} = {⟨ (T + U) (x), y ⟩}_{2}

= {⟨ T (x) + U (x), y ⟩}_{2} = {⟨ T (x), y ⟩}_{2} + {⟨ U (x), y ⟩}_{2}

= {⟨ x, T^{∗} (y) ⟩}_{1} + {⟨ x, U^{∗} (y) ⟩}_{1} = {⟨ x, (T^{∗} + U^{∗}) (y) ⟩}_{1}

for all $x$ and $y$ .

2.

Compute

{⟨ x, {(cT)}^{∗} (y) ⟩}_{1} = {⟨ cT (x), y ⟩}_{2} =

{⟨ T (x), \bar{c} y ⟩}_{2} = {⟨ x, T^{∗} (\bar{c} y) ⟩}_{1} = {⟨ x, \bar{c} T^{∗} (y) ⟩}_{1}

for all $x$ and $y$ .

3.

Let

T

is a mapping on

W

and

U

is a mapping from

V

to

W

. Compute

{⟨ x, {(TU)}^{∗} (y) ⟩}_{1} = {⟨ TU (x), y ⟩}_{2}

= {⟨ U (x), T^{∗} (y) ⟩}_{2} = {⟨ x, U^{∗} T^{∗} (y) ⟩}_{1}

for all $x$ and $y$ .

Let $T$ is a mapping from $V$ to $W$ and $U$ is a mapping on $V$ . Compute

{⟨ x, {(TU)}^{∗} (y) ⟩}_{1} = {⟨ TU (x), y ⟩}_{2}

= {⟨ U (x), T^{∗} (y) ⟩}_{1} = {⟨ x, U^{∗} T^{∗} (y) ⟩}_{1}

for all $x$ and $y$ .

4.

Compute

{⟨ x, T^{∗∗} (y) ⟩}_{2} = {⟨ T^{∗} (x), y ⟩}_{1}

= ⟨ x, T (y) ⟩

for all $x$ and $y$ .

jephianlin

2011-06-27 00:00

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