## 每週問題 July 11, 2016

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.

(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 $6-1=5$