So this here questions got me stuck for a bit:
Determine whether S is a subspace of V, if yes find a basis and its dimension. Let $S$ be the set of all 3x3 symmetric matrices in $V = M_{3x3}$
And here's why: I checked the appropriate axioms and it turns out S is in fact a subspace of V, but my main problem is that before this exercise, to find a basis, I'd find the augmented matrix and solve it (RREF) and then determine the solution, but here I'm not sure if I can.
So I have an approach but not sure which if it works:
If I say for e.g.
Assume the matrix $A=\begin{pmatrix} a & b & c\\ b & d & e\\ c & e & f\\ \end{pmatrix}$
if det(A) $\neq 0$ => linearly independent => the basis exists.
$k_1\begin{pmatrix} a\\ b\\ c\\ \end{pmatrix}$ + $k_2\begin{pmatrix} b\\ d\\ e\\ \end{pmatrix}$ + $k_3\begin{pmatrix} c\\ e\\ f\\ \end{pmatrix} = 0; dim(S) = 3$
Please let me know how to go about this. Thank you.
Hint. The vector space $V$ is $9$-dimensional. How many of the $9$ entries can you specify independently in a $3 \times 3$ matrix and have it be symmetric. Clearly not all $9$.