Basis to subspace

**gralla55** · March 23rd 2010, 03:53 PM

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!

**dwsmith** · May 22nd 2010, 02:51 PM

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: $f=-a-d$

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