Matrix Null Space (Kernel) and Nullity Calculator

The calculator will find the null space (kernel) and the nullity of the given matrix, with steps shown.

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Find the null space of $\left[\begin{array}{ccc}1 & -1 & -1\\2 & -2 & 1\end{array}\right]$.

Solution

The reduced row echelon form of the matrix is $\left[\begin{array}{ccc}1 & -1 & 0\\0 & 0 & 1\end{array}\right]$ (for steps, see rref calculator).

To find the null space, solve the matrix equation $\left[\begin{array}{ccc}1 & -1 & 0\\0 & 0 & 1\end{array}\right]\left[\begin{array}{c}x_{1}\\x_{2}\\x_{3}\end{array}\right] = \left[\begin{array}{c}0\\0\end{array}\right].$

If we take $x_{2} = t$, then $x_{1} = t$, $x_{3} = 0$.

Thus, $\mathbf{\vec{x}} = \left[\begin{array}{c}t\\t\\0\end{array}\right] = \left[\begin{array}{c}1\\1\\0\end{array}\right] t.$

This is the null space.

The nullity of a matrix is the dimension of the basis for the null space.

Thus, the nullity of the matrix is $1$.

The basis for the null space is $\left\{\left[\begin{array}{c}1\\1\\0\end{array}\right]\right\}$A.
The nullity of the matrix is $1$A.