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Math Help - Finding basis of 3x3 matrix

  1. #1
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    Finding basis of 3x3 matrix

    For my homework assignment, I'm supposed to find a basis for the space of 3x3 matrices that have zero row sums and separately for zero row columns. I am having a hard time with this as it seems to me that there are a lot of combinations I have to consider. For the first, it seems like the rows would have to consist of one 0, one 1, and one -1 in different orders... Is there a better way to do this other than brute force?
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Finding basis of 3x3 matrix

    Lets assume we have a 3x3 matrix where the rows and columns sum to zero. Intuitively you know that for the rows to be 0, a + b + c = 0, or a + b = -c, so if you know the values of 2 elements, a and b, you know c.
    \begin{bmatrix} a & b & \underline{\hspace{.10in}} \\ c & d & \underline{\hspace{.10in}} \\ \underline{\hspace{.10in}}& \underline{\hspace{.10in}} & \underline{\hspace{.10in}} \end{bmatrix} so the _ represent values that are automatically determined one you give this matrix concrete values a,b,c,d which can be any real number. So what is the basis for the space of 2x2 matricies over real numbers?

    \begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix},\begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix},\begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix},\begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix}
    if you make each of these 2x2 matricies as the upper 2x2 matrix of a 3x3 matrix, it determines the last row and column

    \begin{bmatrix}1 & 0 & -1\\ 0 & 0 & 0 \\-1 & 0 & 1 \end{bmatrix},\begin{bmatrix}0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & -1 & 1\end{bmatrix},\begin{bmatrix}0 & 0 & 0 \\ 1 & 0 & -1 \\ -1 & 0 & 1 \end{bmatrix},\begin{bmatrix}0 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & -1 & 1 \end{bmatrix}

    thus giving you the basis.
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