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The null space, or kernel of a matrix A is the set of vectors such that

\{ \vec{v} | A \vec{v} = 0 \}

The basis of the null space will be the vectors spanned by the basic solutions of the solution to AX = 0. In terms of linear transformations, the null space of a transformation matrix A is the set of all vectors that are transformed to the zero vector.

For example, suppose we have the matrix and its reduced form

\begin{bmatrix} 1 & 2 & 6 & 2 \\ 0 & 2 & 3 & 0 \\ 1 & 4 & 0 & 4 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & 0 & \frac{8}{3} \\ 0 & 1 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & - \frac{2}{9} \end{bmatrix}

The fourth variable is the non-leading, and can be replaced by the parameter s, yielding a basic solution.

s \begin{bmatrix} \frac{8}{3} \\ \frac{1}{3} \\ - \frac{2}{9} \\ 1\end{bmatrix}

Since s is a free variable, this is the same as

s \begin{bmatrix} 24 \\ 3 \\ -2 \\ -9 \end{bmatrix}

Because the basis has only one vector, the dimension of the null space or nullity is equal to one. By the rank theorem, the dimension of the column and row space will be three.

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