Null Spaces

The null space of a system of equations is the set of solutions for which the dependent variables 'cancel out'. In other words, all values of x such that Ax = 0.

Utility

Defining the null space of a system is useful for defining the 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.

Solving

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

Null spaces will follow a pattern:

• There will be a null space for each free column.
• Populate each vector with the corresponding free column position holding a one, and all other free column positions holding a zero.
• Solve the system of equation given these values and given a right hand side value of 0.

As an example, consider this system:

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

This is eliminated into the following augmented matrix:

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

The free columns are 2 and 4. Therefore, the null space solutions begin like:

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

The first solution can be found by rewriting the first equation from the system (with 0 as the right hand value):

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

Substitute this into the second equation:

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

Substitute this into the first equation again:

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

The first null space solution is:

[-2 1 0 0]

Repeat the process for the second solution, arriving at:

[2 0 -1 1]

Rank

The rank of a matrix is the number of pivots.

If a matrix has only pivot columns, it is said to be full column rank and the only null space is the zero vector (i.e. [0 ... 0]). Ax = b either has one solution or is not solvable.

If a matrix has a pivot in each row, it is said to be full row rank. This only means that Ax = b can be solved for any b.

If a square matrix has full rank, the eliminated matrix is the identity matrix. The only null space is the zero vector; Ax = b can be solved for any b; there is exactly one solution for any given b.

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