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