每週問題 July 11, 2016

判定一個齊次系統的自由變數 (free variable) 的數目。

Suppose that A is the coefficient matrix for a homogeneous system of four equations in six unknowns and suppose that A has at least one nonzero row.
(a) Determine the fewest number of free variables that are possible.
(b) Determine the maximum number of free variables that are possible.

 
參考解答:

(a) 已知 A 是一個 4\times 6 階矩陣。自由變數的數目等於 A 的零空間 (nullspace) 的維數 \dim N(A),稱為零度 (nullity)。根據秩─零度定理,\dim N(A)=6-\hbox{rank}A。因為 \hbox{rank}A\le 4,推得 \dim N(A) 的最小值為 6-4=2

(b) 因為 A 至少有一個非零列 (row),即 \hbox{rank}A\ge 1,推得 \dim N(A) 的最大值為 6-1=5

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