Column space of $$$\left[\begin{array}{ccc}3 & -4 & 2\\1 & 6 & 8\\2 & 7 & 9\end{array}\right]$$$
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Find the column space of $$$\left[\begin{array}{ccc}3 & -4 & 2\\1 & 6 & 8\\2 & 7 & 9\end{array}\right]$$$.
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 column space is a space spanned by the columns of the initial matrix that correspond to the pivot columns of the reduced matrix.
Thus, the column space is $$$\left\{\left[\begin{array}{c}3\\1\\2\end{array}\right], \left[\begin{array}{c}-4\\6\\7\end{array}\right], \left[\begin{array}{c}2\\8\\9\end{array}\right]\right\}.$$$
Answer
The column space of the matrix is $$$\left\{\left[\begin{array}{c}3\\1\\2\end{array}\right], \left[\begin{array}{c}-4\\6\\7\end{array}\right], \left[\begin{array}{c}2\\8\\9\end{array}\right]\right\}.$$$A