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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.

Kode Ch
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  • @DietrichBurde it is the same question, but my approach was not explained there. – Kode Ch May 11 '18 at 15:18
  • Your approach is explained in the duplicate. For example, the matrix with parameter $b$ is $E_{12}+E_{21}$ from the answer there, and for $a$ it is $\frac{1}{2}(2E_{11})=E_{11}$. – Dietrich Burde May 11 '18 at 15:20

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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 $9$.

Ethan Bolker
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