Manuscript XVI-O1-ST@5 · XVI-O1-ST-v5.tex · section · dd45e0d1

Rank two and complete affine ambiguity

Retained exact synthetic examples › Rank two and complete affine ambiguity · lines 1827–1858

Let

M_{1} = 12130241503617048190, L_{1} = - 2 - 2 0 100 0 - 1 0 010001,

Z_{1} = 1 - 2 10 2 - 3 01, y_{1} = (1, 2, 1, 3, 0)^{T}, c_{1} = (1, 0, 0, 0)^{T} .

The original minor on rows $1, 3$ , columns $1, 2$ , is $- 1$ . Columns $2, 4, 5$ of $L_{1}$ form $I_{3}$ ; the last two rows of $Z_{1}$ form $I_{2}$ . Direct multiplication gives

L_{1} M_{1} = 0, L_{1} y_{1} = 0, M_{1} c_{1} = y_{1}, M_{1} Z_{1} = 0.

Hence $rank M_{1} = 2$ and every solution is $c_{1} + Z_{1} t$ .

For

M_{u} = 120240, L_{u} = (- 2 0 1001), y_{u} = (1, 1, 1)^{T},

the rank is one and $L_{u} y_{u} = (- 1, 1)^{T} \neq = 0$ .

Displayed formulas in this passage

unlabeled · line 1830
$M_{1} = 12130241503617048190, L_{1} = - 2 - 2 0 100 0 - 1 0 010001,$
unlabeled · line 1838
$Z_{1} = 1 - 2 10 2 - 3 01, y_{1} = (1, 2, 1, 3, 0)^{T}, c_{1} = (1, 0, 0, 0)^{T} .$
unlabeled · line 1846
$L_{1} M_{1} = 0, L_{1} y_{1} = 0, M_{1} c_{1} = y_{1}, M_{1} Z_{1} = 0.$
unlabeled · line 1852
$M_{u} = 120240, L_{u} = (- 2 0 1001), y_{u} = (1, 1, 1)^{T},$