作者sm008150204 (风切羽狂)
看板Math
标题[线代] v
时间Sun May 8 18:53:05 2011
※ 引述《mqazz1 (无法显示)》之铭言：
: let A be a 3*3 matrix and let x1, x2 and x3 be vector in R^3
: show that if the vectors
: y1 = Ax1
: y2 = Ax2
: y3 = Ax3
: are linearly independent, then the matrix A must be nonsingular and the vector
: x1, x2 and x3 must be linearly independent
: 请问这要怎麽证明呢?
: 我想了两天左右 还不太清楚怎麽证..谢谢
Let y1,y2,y3 be column vetor.
Note that [y1 y2 y3] = [Ax1 Ax2 Ax3]= A[x1 x2 x3] all of them are 3x3 matrix
Since y1,y2,y3 are linearly independent ,so
det[y1 y2 y3] = det(A[x1 x2 x3]) = det(A)det[x1 x2 x3] ≠ 0
thus, det(A) ≠ 0 and det[x1 x2 x3] ≠ 0
which implies that matrix A must be nonsingular and the vector
x1, x2 and x3 must be linearly independent
Done.
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※ 编辑: sm008150204 来自: 218.160.250.142 (05/08 18:55)
1F：→ a88241050 :你怎麽不用回文= = 05/08 19:16
2F：→ sm008150204 :我打到一半突然当机，第一次用暂存档不太会用 抱歉 05/08 21:06
3F：→ sm008150204 :真的是超尴尬的 还是我现在应该复制 然後重回一篇? 05/08 21:08
4F：→ ricestone :把标题改成跟原文一样，程式就会判成相关文章了 05/08 21:11
5F：→ ricestone :顶多不会变成黄色罢了 05/08 21:11
6F：推 mqazz1 :谢谢 05/09 18:47