Exercise 15

Show that every homomorphism from $\mathbb{R}^{1}$ to $\mathbb{R}^{1}$ acts via multiplication by a scalar. Conclude that every nontrivial linear transformation of $\mathbb{R}^{1}$ is an isomorphism. Is that true for transformations of $\mathbb{R}^{2}$? $\mathbb{R}^{n}$?