Show that the set of invertible upper triangular matrices is a submanifold of invertible matrices If $T(n,\mathbb{R}) \subset GL(n,\mathbb{R})$ is the set of upper triangular matrices. I need to show that it's a submanifold of $GL(n,\mathbb{R})$. 
I'm having trouble getting started. I clearly cannot use Regular Value Theorem. In a fix, don't know how to proceed
 A: I finally got a solution. Thought I should post it here
$T(n,\mathbb{R})$ is the set of invertible upper triangular matrices.
Consider $\mathcal{P}\colon GL(n,\mathbb{R})\rightarrow \mathbb{R}^{n(n-1)/2}$, where $\{a_{ij}\}_{ij} \mapsto  (a_{ij})_{i>j}$; $1 \leq i,j \leq n$.
Clearly, $\mathcal{P}^{-1}(0)=T(n,\mathbb{R})$.
We need to verify that $0$ is a regular value of $\mathcal{P}$,
i.e. for any $A \in \mathcal{P}^{-1}(0)$, the map $D\mathcal{P}(A)\colon \text{Mat}(n,\mathbb{R})\rightarrow \mathbb{R}^{n(n-1)/2}$ is surjective.
We can verify that, $\mathcal{P} = \mathcal{P} \circ L_{A^{-1}}$, (where $L_{A^{-1}}$ is the left translation by $A^{-1}$) and by chain rule $D\mathcal{P}(A)=D\mathcal{P}(I)\circ DL_{A^{-1}}$.
So we compute $D\mathcal{P}(I)$. Consider the $\gamma(t)=I + tV$, 
$\mathcal{P}(\gamma(t))=\mathcal{P}(I+tV)=t(v_{ij})_{i>j}$, where $V=\{v_{ij}\}_{ij} \ ; 1 \leq i,j \leq n$
$$\implies D\mathcal{P}(I)(V)=\frac{d}{dt}\mathcal{P}(\gamma(t))\Big|_{t=0}=\frac{d}{dt} t(v_{ij})_{i>j}\Big|_{t=0}=(v_{ij})_{i>j}=\mathcal{Q}(V)$$
Where $\mathcal{Q} \colon \text{Mat}(n,\mathbb{R}) \rightarrow \mathbb{R}^{n(n-1)/2} \ ; \ \{b_{ij}\}_{ij} \mapsto  (b_{ij})_{i>j}$. So, $\mathcal{Q} \vert_{GL(n,\mathbb{R})}=\mathcal{P}$
Hence, $D\mathcal{P}(A)(V)=\mathcal{Q}(A^{-1}V)$.\
Now, to show that, for every $x \in \mathbb{R}^{n(n-1)/2}$, $\exists V \in \text{Mat}(n,\mathbb{R})$ such that $\mathcal{Q}(A^{-1}V)=x$.
We construct $X \in  \text{Mat}(n,\mathbb{R})$ such that $\mathcal{Q}(X)=x$. So, we just take $V=AX$. So, we have shown that $D\mathcal{P}(A)$ is surjective i.e. $\text{rk} (D\mathcal{P}(A)) = n(n-1)/2$
Hence $0$ is a regular value, and therefore, $T(n,\mathbb{R})$ is a submanifold.
Please verify, and suggest edits or/and corrections, if any!
