Matrix Null Space (Kernel) and Nullity Calculator
Find null spaces step by step
The calculator will find the null space (kernel) and the nullity of the given matrix, with steps shown.
Your Input
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 $$$1$$$.
Answer
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.