# Column Space Calculator

The calculator will find the column space of the matrix, with steps shown.

Related calculator: Row Space Calculator

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## Your Input

Find the column space of $\left[\begin{array}{ccc}1 & 2 & 3\\2 & 5 & 4\\1 & 1 & 5\end{array}\right]$.

## Solution

The row echelon form of the matrix is $\left[\begin{array}{ccc}1 & 2 & 3\\0 & 1 & -2\\0 & 0 & 0\end{array}\right]$ (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 $\left\{\left[\begin{array}{c}1\\2\\1\end{array}\right], \left[\begin{array}{c}2\\5\\1\end{array}\right]\right\}$.

## Answer

The column space of the matrix is $\left\{\left[\begin{array}{c}1\\2\\1\end{array}\right], \left[\begin{array}{c}2\\5\\1\end{array}\right]\right\}$A.