let $V$ be the space of real $3\times3$ matrices and let $S\subset V$ be the subspace of symmetric matrices. What is $\dim(S)$?
The dimension of the space of symmetric $3\times 3$ matrices
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1OK I found it. Now I don't see any question words here – 2012-10-10
2 Answers
Hint: Show that the following matrices make a basis for $S$ : $E_1:=\begin{pmatrix} 1 & 0 &0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}; E_2:=\begin{pmatrix} 0 & 1 &0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix};E_3:=\begin{pmatrix} 0 & 0 &1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}\\E_4:=\begin{pmatrix} 0 & 0 &0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix};E_5:=\begin{pmatrix} 0 & 0 &0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix};E_6:=\begin{pmatrix} 0 & 0 &0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} $
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0Nice work, looks pretty ;-) – 2013-03-26
Step 1
There are $9$ entries in a $3\times3$ matrix. Because it is symmetric, the entries are reflected along the diagonal.
Question : What is the maximum number of unique entries? (try writing $3\times3$ symmetric matrices and counting)
Step 2
A standard basis vector of the vector space of $3\times3$ matrices is one that has a $1$ in exactly one entry, and a $0$ in all other entries.
Question : If you take the standard basis vector with a $1$ in entry $(i,j)$ and let it also have a $1$ in entry $(j,i)$. What kind of subspace does this new vector span?
Step 3
Try to combine the previous two steps into a basis for the required subspace.
Question: If there are $n$ vectors in a basis for a subspace, what is the dimension of the subspace? What is the dimension of your required subspace?