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Math Help - Basis to subspace

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    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!
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  2. #2
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    Quote Originally Posted by gralla55 View Post
    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}<br />
a & b & c\\ <br />
b & d & e\\ <br />
c & e & f<br />
\end{bmatrix}
    a+d+f=0
    a=-d-f, d=-a-f, or f=-a-d

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

    Basis:
    \left(\begin{bmatrix}<br />
0 & 1 & 0\\ <br />
1 & 0 & 0\\ <br />
0 & 0 & 0<br />
\end{bmatrix},\begin{bmatrix}<br />
0 & 0 & 1\\ <br />
0 & 0 & 0\\ <br />
1 & 0 & 0<br />
\end{bmatrix},\begin{bmatrix}<br />
-1 & 0 & 0\\ <br />
0 & 1 & 0\\ <br />
0 & 0 & 0<br />
\end{bmatrix},\begin{bmatrix}<br />
0 & 0 & 0\\ <br />
0 & 0 & 1\\ <br />
0 & 1 & 0<br />
\end{bmatrix},\begin{bmatrix}<br />
-1 & 0 & 0\\ <br />
0 & 0 & 0\\ <br />
0 & 0 & 1<br />
\end{bmatrix}\right)

    Case 2: d=-a-f

    Case 3: f=-a-d
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