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[sm008150204 (風切羽狂)]

※ 引述《mqazz1 (無法顯示)》之銘言：
: 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
: 請問這要怎麼證明呢?
: 我想了兩天左右 還不太清楚怎麼證..謝謝

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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※ 編輯: sm008150204     來自: 218.160.250.142      (05/08 18:55)
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