math111_logo Kernel and Null Space

2. Null space

For a linear transformation T: RnRm given by an m by n matrix A, we have a special name for the kernel.

The null space of a matrix A is

nulA = {all solutions of the homogeneous system Ax = 0}.

We remark that the null space of an m by n matrix is a subspace of Rn.

Example The subspace

{(x, y, z): x + y - z = 0, 2x - 3y + z = 0} ⊂ R3

given in an earlier example is in fact the null space of the matrix

[ 1 1 -1 ].
2 -3 1

Moreover, by

[ 1 1 -1 ]
2 -3 1
[ 1 ]
1
1
= [ 1 ]0,
0

we have (1, 1, 1) ∉ nulA, and by

[ 1 1 -1 ]
2 -3 1
[ 2 ]
3
5
= [ 0 ] = 0,
0

we have (2, 3, 5) ∈ nulA.


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