1. ## 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!

2. 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!
$\displaystyle \begin{bmatrix} a & b & c\\ b & d & e\\ c & e & f \end{bmatrix}$
$\displaystyle a+d+f=0$
$\displaystyle a=-d-f$, $\displaystyle d=-a-f$, or $\displaystyle f=-a-d$

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

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

Case 2: $\displaystyle d=-a-f$

Case 3: $\displaystyle f=-a-d$