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.
Contents
Definition
In linear algebra, the null space is the subspace that satisfies Ax = 0. It is notated as N(A).
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
All systems of linear equations have a null space containing the zero vector.
For invertible matrices, the zero vector is the only null space solution.
Solution
Leaving aside the invertible case, the remaining vectors of the null space can be solved for.
Introduction
Consider the below system of equations.
w + 2x + 2y + 2z = 1 2w + 4x + 6y + 8z = 5 3w + 6x + 8y + 10z = 6
This system is rewritten as a linear system and eliminated into:
┌ ┐ ┌ ┐ ┌ ┐
│[1] 2 2 2│ │ w│ │ 1│
│ 0 0 [2] 4│ │ x│ = │ 3│
│ 0 0 0 0│ │ y│ │ 0│
└ ┘ │ z│ └ ┘
└ ┘
Identify Free Columns
For a matrix with n free columns, the null space has n dimensions and has n solutions.
Identify the columns with a pivot in the eliminated form. The remaining columns represent free variables.
Substitute
Because 2 solutions are expected in this example's null space, the solution vectors are pre-populated as:
[w 1 y 0] [w 0 y 1]
The nth vector has a 1 in the nth free variable, and a 0 in all other free variables. The pivot variables are left to vary.
Solve Ax = 0 using these values.
For example, the solution using the 1st vector:
w + 2x + 2y + 2z = 0 w + 2(1) + 2y + 2(0) = 0 w + 2 + 2y = 0 w = -2 - 2y 2w + 4x + 6y + 8z = 0 2w + 4(1) + 6y + 8(0) = 0 2w + 4 + 6y = 0 2w + 4 + 6y = 0 2(-2 - 2y) + 4 + 6y = 0 -4 - 4y + 4 + 6y = 0 2y = 0 y = 0 w = -2 - 2y w = -2 - 2(0) w = -2
Leads to the solution [-2 1 0 0].
Repeat for each vector. The second solution is [2 0 -1 1].