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Atrium > Mathematics, Physics & Philosophy: Incidence matrix

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- CommentRowNumber1.
- CommentAuthorMurat_Aygen
- CommentTimeFeb 14th 2018
- (edited Feb 14th 2018)
- PermaLink
Author: Murat_Aygen
Format: MarkdownItexIf one solves <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mn>1</mn></msub><mo>⟨</mo><msub><mi>a</mi><mn>1</mn></msub><mo>⟩</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>b</mi><mi>m</mi></msub><mo>⟨</mo><msub><mi>a</mi><mi>m</mi></msub><mo>⟩</mo><mo>=</mo><mi>δ</mi><mo>[</mo><mo>∑</mo><mo>∑</mo><msub><mi>x</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>⟨</mo><msub><mi>a</mi><mi>i</mi></msub><mo>,</mo><msub><mi>a</mi><mi>j</mi></msub><mo>⟩</mo><mo>]</mo></math> for x's when b's are given, one ends up with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">η</mi><mover><mi>x</mi><mo>→</mo></mover><mo>=</mo><mover><mi>b</mi><mo>→</mo></mover></math> where \eta is the INCIDENCE MATRIX! Then all proofs about 0-homology groups are just about the ranks of square-submatrices of this matrix. Am I right?

If one solves
$b_{1} ⟨ a_{1} ⟩ + \dots + b_{m} ⟨ a_{m} ⟩ = δ [\sum \sum x_{i, j} ⟨ a_{i}, a_{j} ⟩]$
for x’s when b’s are given, one ends up with
$η \vec{x} = \vec{b}$
where \eta is the INCIDENCE MATRIX! Then all proofs about 0-homology groups are just about the ranks of square-submatrices of this matrix. Am I right?

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