MathJax TeX Test Page
$$\text{1. A is similar to B}$$
$$\text{2. There exists an ordered basis } \mathcal{B} \text{ for } \mathbb{R}^n \text{ such that } [A]_{\mathcal{B}} = B$$
This follows from the
change of basis formula. $[A]_{\mathcal{E}} = \underset{E \leftarrow B}{P}[A]_{\mathcal{B}}\underset{E \leftarrow B}{P}^{-1}$
$$\text{3. There exists a linear transformation T and ordered bases } \mathcal{B}_1 , \mathcal{B}_2 \\ \text{ such that } A = [T]_{\mathcal{B}_1}, B = [T]_{\mathcal{B}_2}$$
Let $[T] = A$. Let $\mathcal{B}_1 = \mathcal{E}, \mathcal{B}_2 = \mathcal{B}$. We can recreate 2.