# Basis to subspace

• Mar 23rd 2010, 03:53 PM
gralla55
Basis to subspace
Let V be the vector space of 3x3 matrices.:

V = M(3x3) and let W be the subspace of symmetrical matrices with thrace 0. The matrix B is also given as:

B =

-4 1 -2
1 9 3
-2 3 -5

How can I find a basis G to the subspace W? Thanks!
• May 22nd 2010, 02:51 PM
dwsmith
Quote:

Originally Posted by gralla55
Let V be the vector space of 3x3 matrices.:

V = M(3x3) and let W be the subspace of symmetrical matrices with thrace 0. The matrix B is also given as:

B =

-4 1 -2
1 9 3
-2 3 -5

How can I find a basis G to the subspace W? Thanks!

$\begin{bmatrix}
a & b & c\\
b & d & e\\
c & e & f
\end{bmatrix}$

$a+d+f=0$
$a=-d-f$, $d=-a-f$, or $f=-a-d$

Case 1:
$a=-d-f$
$b\begin{bmatrix}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}+c\begin{bmatrix}
0 & 0 & 1\\
0 & 0 & 0\\
1 & 0 & 0
\end{bmatrix}+d\begin{bmatrix}
-1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{bmatrix}+e\begin{bmatrix}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{bmatrix}+f\begin{bmatrix}
-1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1
\end{bmatrix}$

Basis:
$\left(\begin{bmatrix}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{bmatrix},\begin{bmatrix}
0 & 0 & 1\\
0 & 0 & 0\\
1 & 0 & 0
\end{bmatrix},\begin{bmatrix}
-1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{bmatrix},\begin{bmatrix}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{bmatrix},\begin{bmatrix}
-1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1
\end{bmatrix}\right)$

Case 2: $d=-a-f$

Case 3: $f=-a-d$