I need help finding the general element or matrix of $V$. Do I need to find the basis of the nullspace and basis of the image to solve this problem?
2026-03-29 22:33:47.1774823627
Find a basis of the space V of all symmetric 3x3 matricies, and thus determine the dimension of V
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A start: For any ordered pair $(i,j)$, where $1\le i\le j\le3$, let $M_{i,j}$ be the matrix that has a $1$ in positions $(i,j)$ and $(j,i)$, and that has $0$ everywhere else.
Show that the $M_{i,j}$ are linearly independent and span our space. It is easy to see that there are $6$ matrices in the collection. The idea generalizes to $n\times n$ symmetric matrices.