Identify the space of all $2 \times 2$ real matrices with $\mathbb{R}^4$ so that the matrix $\left( \begin{array}{cc} a & b\\ c & d\end{array} \right)$
corresponds to $(a, b, c, d)$. Let $\Gamma$ denote the hyperplane in $\mathbb{R}^ 4$ with equation $x_1 +x_2 +x_3 - x_4 = 0$. Does $\Gamma$ intersect $SL(2, \mathbb{R})$ transversely at $I$?�
At $\begin{pmatrix}a&b\\c&d\end{pmatrix}=I$, we have $$\begin{align}\frac{\partial}{\partial a}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=d=1,\\\frac{\partial}{\partial b}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=-c=0, \\\frac{\partial}{\partial c}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=-b=0,\\\frac{\partial}{\partial d}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=a=1.\end{align}$$ hence the tangent space to $SL$ at $I$ is the orthogonal space to $(1,0,0,1)$. On the other hand $\Gamma$ is th eorthogonal space to $(1,1,1,-1)$, hence these two tangent sapces span all of $\mathbb R^4$.