Exercise 6.B.17

Proof. (a) For additivity, suppose $u_1,u_2\in V$. Then, for $v\in V$, we have

For homogeneity, suppose $u\in V$ and $c\in ℝ$. Then, for $v\in V$, we have

(b) If $𝔽=ℂ$, then the homogeneity property of linear maps is not satisfied, because we would have $(\Phi (\mathit{cu}))(v)=\overline{c}(\mathit{\Phi u})(v)$, but $c=\overline{c}$ if and only if $c$ is a real number.

(c) This is the same as the second part in the proof of 6.42. Suppose there are $u_1$ and $u_2$ in $V$ such that $\mathrm{\Psi }{u}_1=\mathrm{\Psi }{u}_2$. Then

for all $v\in V$. Choosing $v=u_1-u_2$ shows that $u_1-u_2=0$ and thus $u_1=u_2$.

(d) From (c), we get that $\mathit{dimnull\Phi }=0$. Thus, from 3.22, we have

However, $\dim <!--\; FUNCTION\; APPLICATION\; -->V=\dim <!--\; FUNCTION\; APPLICATION\; -->V^{\prime }$. This shows that $\Phi$ also surjective. Hence $\Phi$ is invertible, that is, an isomorphism from $V$ to $V^{\prime }$. □