# Equivalent $R$-linear and $C$-linear systems of equations

Abstract : This paper addresses two elementary subjects in linear algebra, intertwining concepts in real and complex numbers, which could be proposed as homework assignments to students learning complex linear algebra. First, given an $R$-linear system of equations with data in complex numbers, necessary and sufficient conditions are given ensuring that there exists a C-linear system of equations of the same size that has the same solution set whatever is the constant term of the original system. The motivation for searching for such an equivalence may be theoretical or based on a numerical efficiency wish. This first result rests on the second contribution of the paper, which claims that, being given an $R$-injective matrix $M\in\mathbb{C}^{m\times(2n)}$ - such a matrix must have more rows than half the number of its columns - one can find a matrix $H\in\mathbb{C}^{n\times m}$ such that $HM$ is also R-injective.
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https://hal.inria.fr/hal-01522334
Contributor : Jean Charles Gilbert <>
Submitted on : Tuesday, May 16, 2017 - 5:38:43 PM
Last modification on : Thursday, April 26, 2018 - 10:27:47 AM
Long-term archiving on: : Friday, August 18, 2017 - 12:20:45 AM

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• HAL Id : hal-01522334, version 2

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Jean Charles Gilbert. Equivalent $R$-linear and $C$-linear systems of equations. [Research Report] INRIA Paris. 2017. ⟨hal-01522334v2⟩

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