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| 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''. | 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. |
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| == Utility == | == Definition == |
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| Defining the null space of a system is useful for defining the '''complete solution'''. | For a system of equations, the '''null space''' is the subspace that satisfies '''''A'''x = 0''. This is useful for finding a [[LinearAlgebra/Solution|complete solution]]. |
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| 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''. | Algebraically, null spaces have an identity property. Given any valid solution to '''''A'''x = b'', ''any'' combination of null spaces can be added to that solution to create another valid solution, because ''b + 0 = b''. |
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| == Solving == | == Zero Vector as a Solution == For all systems of equations, there is a null space. The zero vector is always such a solution. If a matrix '''''A''''' is invertible, the zero vector is the only null space solution. ---- == Solution == Leaving aside the invertible '''''A''''' case, the remaining vectors of the null space can be solved for. === Identify Free Columns === |
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| 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. |
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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| This is eliminated into the following augmented matrix: | The eliminated form of the augmented matrix '''''A''''' looks like: |
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| The free columns are 2 and 4. Therefore, the null space solutions begin like: | Note that the free columns are in positions 2 and 4. === Substitute === Populate each null space vector with a 1 in the corresponding free column position, and a 0 in all other free column positions. Continuing with the same example: |
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| The first solution can be found by rewriting the first equation from the system (with 0 as the right hand value): | For each null space, solve the '''''A'''x = 0'' for the remaining unknowns (''w'' and ''y'' in this example). This can be done with simple algebra. Start with the first null space vector (''[? 1 ? 0]'') and the first unknown (''w''). Any equation from the system can be used, but typically the equation holding the relevant pivot will be easiest to reduce. |
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| Substitute this into the second equation: | Given this solution for ''w'' in terms of ''y'', solve for ''y''. Again, any equation from the system can be used, but typically the equation holding the relevant pivot will be easiest to reduce. |
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| Substitute this into the first equation again: | Given this solution for ''y'', solve for ''w''. |
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| The first null space solution is: | The first null space vector, and therefore the first null space solution, is: |
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| Repeat the process for the second solution, arriving at: | === Repeat === Repeat the substitution process for each null space vector. Continuing with the same example, the complete null space solutions are: |
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| [2 0 -1 1] | [-2 1 0 0] [2 0 -1 1] |
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---- == Full Rank Matrices == If a matrix has no free columns, it is said to be '''full rank''' and the ''only'' null space is the zero vector (i.e. ''[0 ... 0]''). |
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
For a system of equations, the null space is the subspace that satisfies Ax = 0. This is useful for finding a 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.
Zero Vector as a Solution
For all systems of equations, there is a null space. The zero vector is always such a solution.
If a matrix A is invertible, the zero vector is the only null space solution.
Solution
Leaving aside the invertible A case, the remaining vectors of the null space can be solved for.
Identify Free Columns
Given an eliminated matrix, the solution for null space begins with identifying the free columns.
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.
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│ └ ┘
Note that the free columns are in positions 2 and 4.
Substitute
Populate each null space vector with a 1 in the corresponding free column position, and a 0 in all other free column positions. Continuing with the same example:
[? 1 ? 0] [? 0 ? 1]
For each null space, solve the Ax = 0 for the remaining unknowns (w and y in this example). This can be done with simple algebra.
Start with the first null space vector ([? 1 ? 0]) and the first unknown (w). Any equation from the system can be used, but typically the equation holding the relevant pivot will be easiest to reduce.
w + 2x + 2y + 2z = 0 w + 2(1) + 2y + 2(0) = 0 w + 2 + 2y = 0 w = -2 - 2y
Given this solution for w in terms of y, solve for y. Again, any equation from the system can be used, but typically the equation holding the relevant pivot will be easiest to reduce.
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
Given this solution for y, solve for w.
w = -2 - 2y w = -2 - 2(0) w = -2
The first null space vector, and therefore the first null space solution, is:
[-2 1 0 0]
Repeat
Repeat the substitution process for each null space vector.
Continuing with the same example, the complete null space solutions are:
[-2 1 0 0] [2 0 -1 1]