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| = Null Spaces = | = Null Space = |
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| 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]]. | In linear algebra, the '''null space''' is the subspace that satisfies '''''A'''x = 0''. It is notated as ''N('''A''')''. Null spaces are composed of null space vectors. There is always a '''zero vector''' in the null space. If a matrix is [[LinearAlgebra/Invertibility|invertible]] however, the zero vector is the ''only'' null space vector. |
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| == Zero Vector as a Solution == | == Solutions == |
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| For all systems of equations, there is a null space. The zero vector is always such a solution. | Leaving aside the invertible case, the non-zero vectors of the null space can be solved for. |
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| 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 [[LinearAlgebra/Elimination|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: |
Consider the below system of equations. |
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| w + 2x + 2y + 2z = a 2w + 4x + 6y + 8z = b 3w + 6x + 8y + 10z = c |
w + 2x + 2y + 2z = 1 2w + 4x + 6y + 8z = 5 3w + 6x + 8y + 10z = 6 |
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| The eliminated form of the augmented matrix '''''A''''' looks like: | This system is formulated as a matrix and eliminated into: |
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| ┌ ┐ │ [1] 2 2 2 a│ │ 0 0 [2] 4 b-2a│ │ 0 0 0 0 c-b-a│ └ ┘ |
┌ ┐ ┌ ┐ ┌ ┐ │[1] 2 2 2│ │ w│ │ 1│ │ 0 0 [2] 4│ │ x│ = │ 3│ │ 0 0 0 0│ │ y│ │ 0│ └ ┘ │ z│ └ ┘ └ ┘ |
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| Note that the free columns are in positions 2 and 4. | The number of vectors in the null space match the number of free variables. In this case, there are 2. |
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| 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]'' |
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| === 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 '''''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. |
Solve '''''A'''x = 0'' using these values. For example, substituting in the first vector's values gives... |
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| w = -2 - 2y | |
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| 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. {{{ |
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| 2w + 4 + 6y = 0 | |
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| }}} | |
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| Given this solution for ''y'', solve for ''w''. {{{ |
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| The first null space vector, and therefore the first null space solution, is: | ...a null space vector in ''[-2 1 0 0]''. |
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| {{{ [-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] }}} |
The second null space vector is ''[2 0 -1 1]''. |
Null Space
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
In linear algebra, the null space is the subspace that satisfies Ax = 0. It is notated as N(A).
Null spaces are composed of null space vectors. There is always a zero vector in the null space. If a matrix is invertible however, the zero vector is the only null space vector.
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.
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│ └ ┘
└ ┘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].
The second null space vector is [2 0 -1 1].