Null Spaces

A null space is a particular category of subspaces. The null space of a system of equations is the set of solutions for which the dependent variables cancel out.

Definition

For a system of equations, the null space is the subspace that satisfies Ax = 0. This is useful for finding a complete solution.

Algebraically, null spaces have an identity property. Given any valid solution to Ax = b, any combination of null spaces can be added to that solution to create another valid solution, because b + 0 = b.

Zero Vector as a Solution

For all systems of equations, there is a null space. The zero vector is always such a solution.

If a matrix A is invertible, the zero vector is the only null space solution.

Solution

Leaving aside the invertible A case, the remaining vectors of the null space can be solved for.

Identify Free Columns

Given an eliminated matrix, the solution for null space begins with identifying the free columns.

There will be a null space vector for each free column. In other words, given a matrix A with n free columns, the null space of A (sometimes notated as N(A)) has n dimensions.

As an example, consider this system:

w + 2x + 2y + 2z = a
2w + 4x + 6y + 8z = b
3w + 6x + 8y + 10z = c

The eliminated form of the augmented matrix A looks like:

┌                  ┐
│ [1] 2  2  2     a│
│  0  0 [2] 4  b-2a│
│  0  0  0  0 c-b-a│
└                  ┘

Note that the free columns are in positions 2 and 4.

Substitute

Populate each null space vector with a 1 in the corresponding free column position, and a 0 in all other free column positions. Continuing with the same example:

[? 1 ? 0]
[? 0 ? 1]

For each null space, solve the Ax = 0 for the remaining unknowns (w and y in this example). This can be done with simple algebra.

Start with the first null space vector ([? 1 ? 0]) and the first unknown (w). Any equation from the system can be used, but typically the equation holding the relevant pivot will be easiest to reduce.

w + 2x + 2y + 2z = 0
w + 2(1) + 2y + 2(0) = 0
w + 2 + 2y = 0

w = -2 - 2y

Given this solution for w in terms of y, solve for y. Again, any equation from the system can be used, but typically the equation holding the relevant pivot will be easiest to reduce.

2w + 4x + 6y + 8z = 0
2w + 4(1) + 6y + 8(0) = 0
2w + 4 + 6y = 0

2(-2 - 2y) + 4 + 6y = 0
-4 - 4y + 4 + 6y = 0
2y = 0
y = 0

Given this solution for y, solve for w.

w = -2 - 2y
w = -2 - 2(0)
w = -2

The first null space vector, and therefore the first null space solution, is:

[-2 1 0 0]

Repeat

Repeat the substitution process for each null space vector.

Continuing with the same example, the complete null space solutions are:

[-2 1  0 0]
[2  0 -1 1]

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