Are these subspaces connected or not?

Question

This question was asked in a masters entance exam of an institute for which I am preparing and I was unable to solve it .

Which of the following spaces are connected ?

1. The set of upper triangular matrices as a subspace of $M_{n}(\mathbb{R})$ .

2.The set of invertible diagonal as a subspace of $M_{n}(\mathbb{R})$ .

Although I have studied topology from Wayne Patty but this type of questions were not there and our instructer was very unprofessional . So , I was a bit stumped upon seeing connectedness of matrices and would really appreciate a detailed answer.

Answer 1

HINT: I’m assuming that the topology on $M_n(\Bbb R)$ is the one that makes it homeomorphic to $\Bbb R^{n^2}$.

Since there are $n(n-1)/2$ entries below the diagonal of an $n\times n$ matrix, the first question is like asking whether

$$A=\left\{\langle x_1,\ldots,x_{n^2}\rangle:x_1=\ldots=x_{n(n-1)/2}=0\right\}$$

is connected in $\Bbb R^{n^2}$. That’s rather like looking at the $yz$-plane or the $z$-axis in $\Bbb R^3$, and if you think about it, you should see that $A$ is homeomorphic to $\Bbb R^m$ for an $m$ that you can compute in terms of $n$.

For the second one, try showing that the set of diagonal matrices whose diagonal entries are all positive is both closed and open in the set of invertible diagonal matrices.

Answer 2

For the first one, see Marso's answer in this thread, where it is shown that the first space is path-connected.

For the second one, see Alex's answer in this thread. As Brian pointed out in the comments below, the answer has been written over $\mathbb{C}$, and unfortunately, it does not work over $\mathbb{R}$.