Row space of $$$\left[\begin{array}{ccc}3 & 1 & 2\\-4 & 6 & 7\\2 & 8 & 9\end{array}\right]$$$
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Find the row space of $$$\left[\begin{array}{ccc}3 & 1 & 2\\-4 & 6 & 7\\2 & 8 & 9\end{array}\right]$$$.
Solution
The reduced row echelon form of the matrix is $$$\left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right]$$$ (for steps, see rref calculator).
The row space is a space spanned by the nonzero rows of the reduced matrix.
Thus, the row space is $$$\left\{\left[\begin{array}{c}1\\0\\0\end{array}\right], \left[\begin{array}{c}0\\1\\0\end{array}\right], \left[\begin{array}{c}0\\0\\1\end{array}\right]\right\}.$$$
Answer
The row space of the matrix is $$$\left\{\left[\begin{array}{c}1\\0\\0\end{array}\right], \left[\begin{array}{c}0\\1\\0\end{array}\right], \left[\begin{array}{c}0\\0\\1\end{array}\right]\right\}.$$$A