# Superposition Principle

*Superposition principle* states that a *linear combination* of solutions to a linear
equation is again a solution. In general, if V is a linear operator from one vector space to another and we have that Ax_{1} = f_{1} and Ax_{2} = f_{2} we may assert that _{1}x_{1} + a_{2}x_{2}) = a_{1}f_{1} + a_{2}f_{2}.

a_{1}x_{1} + a_{2}x_{2}

where a_{1} and a_{2} are constants, is known as a *linear combination* of vectors x_{1} and x_{2}. (Of course, we may consider more vectors and a longer combination.) It appears that the Superposition Principle is nothing but a truism. This is to a great extent true. Which is not to say that it can't be put to a good use.

As an example, let's consider the Magic Squares puzzle. The fine theory behind the puzzle does not provide a convenient clue to its solution. Just to remind, we enumerated squares of a 3x3 board left to right top to bottom which led to a 9 dimensional boolean space. v_{o} and v_{T} were the original and the target configurations of the checkboxes. And the solution is given by _{T} - v_{o} = sV,_{o} = v_{T}

- v
_{o}and v_{T}differ in the upper left corner. The solution is(1, 1, 1, 1, 1, 0, 1, 0, 0) or using 1 for a checked box and 0 for an unchecked one we have the following configuration:

1 1 1 1 1 0 1 0 0 Let's call it s

_{1}. For other corners, the configuration must be rotated correspondingly. Denote them as s_{3}, s_{7}and s_{9}according to our enumeration rule. - v
_{o}and v_{T}differ in the middle box of the upper row. The solution s_{2}is then given by(0, 1, 0, 1, 0, 1, 1, 0, 1) or

0 1 0 1 0 1 1 0 1 Solutions s

_{4}, s_{6}, and s_{8}are obtained by rotation. - v
_{o}and v_{T}differ in the middle square.s or_{5}= (0, 1, 0, 1, 1, 1, 0, 1, 0)0 1 0 1 1 1 0 1 0

Now the interesting part. Assume, for example that v_{o} and v_{T} differ at the two upper corners.
Then the solution to the puzzle is a combination (actually a simple boolean sum or a componentwise XOR of
s_{1} and s_{3}. Which is none other than

0 | 0 | 0 |

1 | 0 | 1 |

1 | 0 | 1 |

Thus one way to solve the puzzle is to identify the boxes where the given and target configurations differ and XOR the corresponding solutions s_{i} to partial problems.

Another fine example of application of the superposition principle comes from the study of the Stern-Brocot trees.

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