Find the dimension of the system with equations and variables I'm studying dimension of the group now. But I have some trouble with it.
Consider the dimension of 2 by 2 orthogonal group, $O(2)=\{A \in M(2,2,R): AA^T=I\}$
Let $A = \begin{bmatrix}
a & b \\
c & d 
\end{bmatrix}$.
Then by some calculation, we get three equations $a^2+b^2=1 , c^2+d^2=1, ac+bd=0$.
So, can I say that 'there was four free variable, and three equations, now it's dimesion is one although equations are not linear equation'?
Thank you

Yes and no.
You need to show that your system of equations does not include "repetition".
Suppose your conditions $A \in G \iff \phi(A)=0$, where $\phi : M_n(\mathbb{R}) \rightarrow \mathbb{R}^p$ (i.e. "there are $p$ condition equations).
In order to be able to conclude that the dimension of $G$ as a Lie group is $n^2 - p$, you need to show that $\phi$ is locally surjective. This translates to a condition on the rank of the differential of $\phi$.
More specifically, your conditions could be rephrased as $\phi(A) = AA^T-I = 0$, which is $n^2$ equations (in your case, n=2). However, only $n(n+1)/2$ of those are "free equations", namely the rank of the differential of $\phi$ is only  $n(n+1)/2$, and the dimension of $O(n)$ is $n(n-1)/2$.
