Research Scholarships Screening Test Saturday, January 19, 2013 Q. 3.5

Question

Notation:

\( \mathbb{R}^n \) (respectively, \( \mathbb{C}^n \)) denotes the \( n \)-dimensional Euclidean space over \( \mathbb{R} \) (respectively, over \( \mathbb{C} \)), and is assumed to be endowed with its `usual' topology.  
\( M_n(\mathbb{R}) \) (respectively, \( M_n(\mathbb{C}) \)) will denote the set of all \( n \times n \) matrices with entries from \( \mathbb{R} \) (respectively, \( \mathbb{C} \)) and is identified with \( \mathbb{R}^{n^2} \) (respectively, \( \mathbb{C}^{n^2} \)) when considered as a topological space.

Let
\[
S = \{ A \in M_n(\mathbb{R}) : \operatorname{tr}(A) = 0 \}.
\]
Which of the following statements are true?

    a. \( S \) is nowhere dense in \( M_n(\mathbb{R}) \).
    b. \( S \) is connected in \( M_n(\mathbb{R}) \).
    c. \( S \) is compact in \( M_n(\mathbb{R}) \).

 

Answer 1

Our universe is $M_n(\mathbb{R})$, the space of all $n \times n$ matrices, which is identified with $\mathbb{R}^{n^2}$—a big, flat, Euclidean space of dimension $n^2$.

The condition
\[
\operatorname{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} = 0
\]
defines a linear equation in the entries of $A$.

In $\mathbb{R}^3$, an equation like $x + y + z = 0$ defines a flat plane passing through the origin.

Similarly, in our $\mathbb{R}^{n^2}$ space, the equation $a_{11} + \cdots + a_{nn} = 0$ defines a flat hyperplane passing through the origin.

Therefore, $S$ is a giant, flat hyperplane living inside the larger space $M_n(\mathbb{R})$.

Now just remember:

1.  Every hyperplane in a vector space of dimension $> 1$ is a classic example of a nowhere dense set.
2.  Every hyperplane is a prime example of a connected set.
3.  Every hyperplane in a vector space (of positive dimension) is non-compact because it is unbounded.

Hence, statements (a) and (b) are TRUE, and statement (c) is FALSE.