0
$\begingroup$

Let $V = M_{22}(\Bbb R)$ be the set of $2 \times 2$ matrices with real entries. If A = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} is an element of $V$ , then $\mathrm{tr}\, A = a + d$ is called the trace of $A$.

$W = \{A \in V \mid A^T = - A\}$, $U = \{A \in V \mid \mathrm{tr}\, A =0\}$ are subspaces of $V$.

I have already proved that $U$, $W$ are subspaces of $V$ and that $W$ is a subspace of $U$, but how do I go about proving that they do not coincide?

  • 3
    $W$ is not a subspace of $A$. Why would a symmetric matrix have a zero trace? – 2017-01-22
  • 0
    Maybe you mean A^T=-A$? – 2017-01-22
  • 0
    Yeah, I meant $A^T$= - A – 2017-01-22
  • 0
    @HagenvonEitzen – 2017-01-22

2 Answers 2

0

$W$ is not a subspace of $U$. The matrix

$$A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \\ \end{pmatrix}$$

is symmetric and we have $\mathrm{tr}\, A = 2$.

Therefore $U$ and $W$ do not coincide; for another example, find a non-symmetric matric with trace $0$. This is straightforward.

  • 0
    Or, as $W$ is not even a subspace of $U$, we already know they do not coincide ... – 2017-01-22
  • 0
    Yes, indeed that is enough. – 2017-01-22
  • 0
    The question has changed: $W$ consists of the antisymmetric matrices. – 2017-01-22
0

If $A^T=-A$ then for all diagonal elements, we find $a_{ii}=-a_{ii}$, i.e., $a_{ii=0}$ and in particular $\operatorname{tr} A=\sum_i a_{ii}=0$. Thus $W$ is a subspace of $U$.

To show that it is a proper subspace, it suffices to exhibit a single matrix that trace $=0$, but not all diagonal elements $=0$. can you see a way to have $a_{11}+a_{22}=0$ with $a_{11}\ne 0$?