Question

线性代数简单一题

线性代数简单一题后面字是，求a3a4使得a1a2a3a4线性无关，带过程

Answer 1

设 a3 = (p, q, s, t)^T, a4 = (u, v, w, x)^T
则 A = (a1, a2, a3, a4) =
[ 1 1 p u]
[ 1 2 q v]
[-1 0 s w]
[-1 3 t x]
初等行变换为
[ 1 1 p u]
[ 0 1 q-p v-u]
[ 0 1 s+p w+u]
[ 0 4 t+p x+u]
初等行变换为
[ 1 1 p u]
[ 0 1 q-p v-u]
[ 0 0 s+2p-q w+2u-v]
[ 0 0 t+5p-4q x+5u-4v]
答案不唯一，不妨设 s = 1，p = 1， q = 2， t = 3；
即 A 初等行变换为
[ 1 1 1 u]
[ 0 1 1 v-u]
[ 0 0 1 w+2u-v]
[ 0 0 0 x+5u-4v]
x+5u-4v ≠ 0，不妨设 u = 1，v = 1， w = 0， x = 0；
a3 = (1, 2, 1, 3)^T, a4 = (1, 1, 0, 0)^T 即为满足要求的一组解。