Null Space
A null space or kernel is a subspace spanned by the vectors that map to the zero vector.
Definition
Because zero vectors have an identity property (i.e., any linear combination of them can be added to arrive at the exact same values), they have an important role in identifying the general solution to a system of equations.
Therefore the vectors which map to a zero vector must be identified, as through Ax = 0. Identifying all independent vectors gives a basis for spanning a subspace called the null space or the kernel. It is notated as N(A). The vectors themselves are then called null space vectors.
The number of dimensions spanned by the null space vectors is called nullity, and it has a direct relation to rank. For any matrix A of size m x n (i.e., there are n columns), rank(A) + nullity(A) = n.
Note that Elimination does not change the null space of a matrix. Therefore elimination is a strategy for identifying both rank and nullity.
There is always a trivial component of the null space: the zero vector itself. If a matrix is invertible, this is the only null space vector.
Solutions
Leaving aside the invertible case, the non-zero vectors of the null space can be solved for.
Consider the below system of equations.
w + 2x + 2y + 2z = 1 2w + 4x + 6y + 8z = 5 3w + 6x + 8y + 10z = 6
This system is formulated as a matrix and eliminated into:
┌ ┐ ┌ ┐ ┌ ┐
│[1] 2 2 2│ │ w│ │ 1│
│ 0 0 [2] 4│ │ x│ = │ 3│
│ 0 0 0 0│ │ y│ │ 0│
└ ┘ │ z│ └ ┘
└ ┘
Back-Substitution Procedure
The number of vectors in the null space match the number of free variables. In this case, there are 2.
The null space vectors can be 'prepopulated' or 'templated' by allowing pivot variables to vary and alternating which free variable is set to 0 or 1. In this case, they are:
[w 1 y 0]
[w 0 y 1]
Solve Ax = 0 using these values. For example, substituting in the first vector's values gives...
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
...a null space vector in [-2 1 0 0].
Linear Combinations Procedure
Consider combinations of the rows that sum to a zero vector. The first null space vector found was [-2 1 0 0], corresponding to -2 times the first column and 1 times the second column (and zero times the other two). It should be immediately clear that -2*[1 0 0] + 1*[2 0 0] = [0 0 0].
Following this framework, the second null space vector is [2 0 -2 1] because 2*[1 0 0] + -2*[2 2 0] + 1*[2 4 0] = [0 0 0]