Re: [理工] 線代
※ 引述《fifisuccess (fifi)》之銘言：
: 是非題
: (1)在任何一向量空間中,若ax=bx 則a=b
: (2)在任何一向量空間中,若ax=ay 則x=y
: (3)若S為向量空間V的子集,則span(S)等於V中所有包含S之子空間的交集
: (1)F(2)T(3)T
: (4)所有n*n矩陣滿足其跡數等於零所成集合是M(F)的子空間W,求W的維度為何?!
: (4)n^2-1
nxn nxn
W=span{A€M |trace(A)=0} = span{A€M |Σaii = 0}
nxn n-1
=span{A€M |ann = Σ -aii}
i=1
= span{(Aij|i,j=1,...,n)\Ann,Aij代表只有aij不為0}
我看我寫清楚一點好了
[0 0 0 ... 0 ... 0]
[0 0 0 ... 0 ... 0]
[. . . . .]
[. . . . .]
Aij =[0 0 0 ... k ... 0]第i列
[. . . . .]
[. . . . .]
[0 0 0 ... 0 ... 0]
第
j
行
所以W = span{A11,A12,A13,...,A21,A22,A23,...,An1,An2,...An(n-1),Ann}
n-1
但是Ann = Σ -Aii
i=1
所以W = span{A11,A12,A13,...,A21,A22,A23,...,An1,An2,...An(n-1)}
dim W =n^2 -1
: 謝謝^^
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◆ From: 219.70.144.153
※ 編輯: ILzi 來自: 219.70.144.153 (09/29 00:12)
→
09/29 00:22, , 1F
09/29 00:22, 1F
→
09/29 00:25, , 2F
09/29 00:25, 2F
推
09/29 15:44, , 3F
09/29 15:44, 3F
→
09/29 15:45, , 4F
09/29 15:45, 4F
→
09/29 15:53, , 5F
09/29 15:53, 5F
就是說(Aij|i，j=1,..,n)總共有A11,A12,...,Ann這n^2種狀況，然後扣掉Ann這個不算
※ 編輯: ILzi 來自: 219.70.144.153 (09/29 19:23)
※ 編輯: ILzi 來自: 219.70.144.153 (09/29 19:30)
推
09/30 00:27, , 6F
09/30 00:27, 6F
→
09/30 00:28, , 7F
09/30 00:28, 7F
→
09/30 00:31, , 8F
09/30 00:31, 8F
→
09/30 00:31, , 9F
09/30 00:31, 9F
→
09/30 00:33, , 10F
09/30 00:33, 10F
→
09/30 00:35, , 11F
09/30 00:35, 11F
→
09/30 00:36, , 12F
09/30 00:36, 12F
→
09/30 00:38, , 13F
09/30 00:38, 13F
→
09/30 00:40, , 14F
09/30 00:40, 14F
→
09/30 00:41, , 15F
09/30 00:41, 15F
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