Exercise 2.6.2

In these question we should check whether it’s linear and whether its domain and codomain are $V$ and $F$ respectively.

1.

Yes. We may check that

$$f(p(x)+\mathit{cq}(x))=2p^{\prime }(0)+2c{p}^{\prime }(0)+p^{″}(1)+c{q}^{″}(1)$$

$$2p^{\prime }(0)+p^{″}(1)+c(2p^{\prime }(0)+q^{″}(1))=f(p(x))+\mathit{cf}(q(x)).$$

2.

No. It’s codomain should be the field.

3.

Yes. We may check that

$$\mathrm{tr}(A+\mathit{cB})=\sum _{i=1}^n{(A+\mathit{cB})}_{\mathit{ii}}$$

$$=\sum _{i=1}^n{A}_{\mathit{ii}}+c{B}_{\mathit{ii}}=\mathrm{tr}(A)+c\mathrm{tr}(B).$$

4.

No. It’s not linear.

5.

Yes. We may check that

$$f(p(x)+\mathit{cq}(x))={\int \; <!--\; nolimits\; -->}_0^1(p(t)+\mathit{cq}(t))\mathit{dt}$$

$$={\int \; <!--\; nolimits\; -->}_0^{1}p(t)\mathit{dt}+c{\int \; <!--\; nolimits\; -->}_0^{1}q(t)\mathit{dt}=f(p(x))+\mathit{cf}(q(x)).$$

6.

Yes. We may check that

$$f(A+\mathit{cB})={(A+\mathit{cB})}_{11}=A_{11}+c{B}_{11}=f(A)+\mathit{cf}(B).$$