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.
Contents
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.