[線代] linearly independent

看板Math作者sm008150204 (風切羽狂)時間14年前 (2011/05/08 18:53)推噓1(1推 0噓 5→)

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※ 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. -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 218.160.250.142 ※ 編輯: sm008150204 來自: 218.160.250.142 (05/08 18:55)

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