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express an affine space as linear system of equations

asked 2019-06-04 13:17:43 +0200

Michael_Huang gravatar image

Given an affine space, for example {25,27,29,31} in $\mathbb{F}_2^4$, how can I find the equations to express the space as $Ax=b$. I believe there must be some commands in sage can help me. But I can not find them.

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answered 2019-06-04 17:23:47 +0200

tmonteil gravatar image

updated 2019-06-04 17:26:17 +0200

Is it homework ? Here are some hints:

  • if you remove the first element to all elements of an affine space, you get a vector space (see VectorSpace and its vector_space_span method)
  • this vector space has a basis
  • turn this basis into a matrix M (by columns)
  • the lines of the matrix A you are looking for are a basis of the right kernel of M (write the things down on a paper to get convinced)
  • you get the b by plugging any element of your affine space in the equation
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Asked: 2019-06-04 13:17:43 +0200

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Last updated: Jun 04 '19