Question

线性代数问题

线性代数问题如图 求过程

Answer 1

A = (a1, a2, a3, a4, a) =
[1 -1 3 -2 4]
[1 -3 2 6 1]
[1 5 -1 10 6]
[3 1 p+2 p 10]
初等行变换为
[1 -1 3 -2 4]
[0 -2 -1 8 -3]
[0 6 -4 12 2]
[0 4 p-7 p+6 -2]
初等行变换为
[1 0 7/2 -6 11/2]
[0 1 1/2 -4 3/2]
[0 0 -7 36 -7]
[0 0 p-9 p+22 -8]
初等行变换为
[1 0 0 12 2]
[0 1 0 -10/7 1]
[0 0 1 -36/7 1]
[0 0 0 (43p-170)/7 1-p]
(1) -7/(p-9) ≠ 36/(p+22), P ≠ 170/43 时，
a1, a2, a3, a4 线性无关，此时进一步初等行变换为
[1 0 0 0 (170p-424)/(43p-170)]
[0 1 0 0 (33p-160)/(43p-170)]
[0 0 1 0 (7p-134)/(43p-170)]
[0 0 0 1 7(1-p)/(43p-170)]
a = [(170p-424)a1+(33p-160)a2+(7p-134)a3+7(1-p)a4]/(43p-170);
(2) P = 170/43 时， a1, a2, a3, a4 线性相关。
r(a1, a2, a3, a4) = 3, a1, a2, a3 是一个极大线性无关组。