I am trying to solve the following exercise,
What is the smallest subspace of 3 by 3 matrices that contains all symmetric matrices and all lower triangular matrices? Moreover, what is the largest subspace that is contained in both of those subspaces?
I would be very grateful if you could help me to understand in any way, either with a hint or an analytical explanation, to the following:
- Is there a standard definition of a subspace so as to describe mathematically the initial space of $3 \times 3$ matrices and the contained subspaces that formed by the lower triangular and symmetric matrices?
- Given that the space of $3 \times 3$ matrices contains all symmetric matrices and all lower triangular matrices,
- Can we write this literal explanation in mathematical terms using matrices?
- What is the intuitive explanation?
- Can we prove that the space of $3 \times 3$ matrices is closed under addition and scalar multiplication?
- What methodology should we follow so as to compute the smallest and largest $3\times3$ subspace?
- Finally, is there a geometrical explanation to this problem?
If that matters, I would like to note that this is not an exercise for a course but for a personal pursue:)