Null space of [2200]\left[\begin{array}{cc}-2 & 2\\0 & 0\end{array}\right]

The calculator will find the null space of the 22x22 matrix [2200]\left[\begin{array}{cc}-2 & 2\\0 & 0\end{array}\right], with steps shown.
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Your Input

Find the null space of [2200]\left[\begin{array}{cc}-2 & 2\\0 & 0\end{array}\right].

Solution

The reduced row echelon form of the matrix is [1100]\left[\begin{array}{cc}1 & -1\\0 & 0\end{array}\right] (for steps, see rref calculator).

To find the null space, solve the matrix equation [1100][x1x2]=[00].\left[\begin{array}{cc}1 & -1\\0 & 0\end{array}\right]\left[\begin{array}{c}x_{1}\\x_{2}\end{array}\right] = \left[\begin{array}{c}0\\0\end{array}\right].

If we take x2=tx_{2} = t, then x1=tx_{1} = t.

Thus, x=[tt]=[11]t.\mathbf{\vec{x}} = \left[\begin{array}{c}t\\t\end{array}\right] = \left[\begin{array}{c}1\\1\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 11.

Answer

The basis for the null space is {[11]}\left\{\left[\begin{array}{c}1\\1\end{array}\right]\right\}A.

The nullity of the matrix is 11A.