Automorphisms of Affine space $\mathbb R^2$ under additive structure is $GL(2,R)$

Question

I am stuck at proving how linear transformations correspond to automorphisms. I thought of trying to show a one to one correspondence between $\operatorname{Aut}(\mathbb R^2,+)$ and $GL$($2$,$R$). I know that $2*2$ invertible matrices form one to one correspondence to linear transformation maps from $R^2$ to itself but I have no rigorous mathematical prove to conclude that these linear transformations form one to one correspondence with $\operatorname{Aut}(\mathbb R^2,+)$. Is it right if I showed Linear transformations are forming bijective homomorphism (under additive structure)? Any help or hints are appreciated.

Answer 1

I will denote $\operatorname{Aut}(\mathbb{R}^2)$ to be the group of bijective real-linear maps $\mathbb{R}^2 \to \mathbb{R}^2$. We construct a bijection $\phi: \operatorname{Aut}(\mathbb{R}^2) \to GL_2(\mathbb{R})$ as follows. Let $e_1, e_2$ be the standard basis of $\mathbb{R}^2$. If $T \in \operatorname{Aut}(\mathbb{R}^2)$, we define the matrix $\phi(T)$ to be $$\phi(T) = \begin{bmatrix} Te_1 & Te_2 \end{bmatrix}.$$

Now $\phi$ is injective. If $\phi(T) = \phi(T')$, then $Te_1 = T'e_1$ and $Te_2 = T'e_2$ so that $T = T'$ on all vectors in $\mathbb{R}^2$. Similarly, $\phi$ is surjective. If $M \in GL_2(\mathbb{R}^2)$, then the map $T_M: \mathbb{R}^2 \to \mathbb{R}^2$ given by $v \mapsto Mv$ is a linear automorphism of $\mathbb{R}^2$ so it is in $\operatorname{Aut}(\mathbb{R}^2)$ and $\phi(T_M) = M$. Hence, $\phi$ is injective and surjective, completing the proof.

I left some gaps but they should make for worthwhile exercises in getting used to this concept. The above construction holds for any finite-dimensional vector space and $\phi$ is not only a bijection but an isomorphism of groups. For more details on this, any linear algebra text should cover the matrix associated to a linear transformation. For example, see Friedberg, Insel, and Spence sections 2.2 and 2.3.