smallest subspace of $3\times 3$ matrices that contains all symmetric matrices and all lower triangular matrices What is the smallest subspace of $3\times 3$ matrices that contains all symmetric matrices and all lower triangular matrices. What is the largest subspace that is contained in both of these subspaces.
My approach: I guess that collection of all diagonal $3\times 3$ matrices will be answer of both....I am not sure... 
 A: The smallest subspace containing two given subspaces is their sum i.e. $S = U+V$.
A symmetric matrix will be $a_{ij} = a_{ji}$ for all $i,j$. A lower triangular matrix is one where $a_{ij} = 0$ if $i < j$.
First we will take care of what contains both:

Theorem: Any matrix can be written as the sum of a symmetric matrix and a lower triangular matrix.

Proof: Suppose $A$ is a matrix. Then, consider the matrix $B$ given by $B_{ij} = A_{ij}$ for $i \leq j$, such that $B$ is symmetric (which takes care of $i>j$ case). Now, for the matrix $A-B$, note that $(A-B)_{ij} = A_{ij}-B_{ij} = 0$ for $i \leq j$. Hence,  $A-B$ is a lower triangular matrix. So $A = B + (A-B)$, is the sum of a symmetric and lower triangular matrix.
Hence, the answer is the entire space of matrices.
The intersection of the two subspaces consists of matrices which are lower triangular and symmetric. Because the zeros on top get reflected on the bottom due to symmetry, such a matrix can only be a diagonal matrix. It is easy to see that every diagonal matrix is symmetric and lower triangular. Hence, the answer to this is the space of diagonal matrices.
