Exercise 6.3.3

Use the definition and that skill used in the previous exercises.

1.

By definition we have

$$⟨(a,b),T^{\text{∗}}(x)⟩=⟨(2a+b,a-3b),(3,5)⟩=11a-12b.$$

We may observe that $T^{\text{∗}}(x)=(11,-12)$.

2.

By definition we have

$$⟨(z_1,z_2),T^{\text{∗}}(x)⟩=⟨(2z_1+i{z}_2,(1-i)z_1),(3-i,1+2i)⟩$$

$$=(5-i)z_1+(-1+3i)z_2.$$

We may observe that $T^{\text{∗}}(x)=(5+i,-1-3i)$.

3.

By definition we have

$$⟨\mathit{at}+b,T^{\text{∗}}(x)⟩=⟨3\mathit{at}+(a+3b),4-2t⟩$$

$$={\int \; <!--\; nolimits\; -->}_{-1}^1-6a{t}^2+(-2a+6b)t+(4a+12b)\mathit{dt}$$

We may observe that $T^{\text{∗}}(x)=6t+12$.