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 * There will be a null space for each free column.  * 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.
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From this point, the null space of ''A'' (sometimes notated as ''N(A)'') can be computed. From this point, the null space of ''A'' can be computed.

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 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.

  • 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

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│
└                  ┘

From this point, the null space of A can be computed.

Note that the free columns are in positions 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]


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LinearAlgebra/NullSpaces (last edited 2024-02-06 03:28:16 by DominicRicottone)