[理工] [線代]向量空間

[dkcheng (電磁霸主)]

Let A，B and C be three matrices，and AB=C. Which of the followings are true ?


(A) The row space of C is a subspace of the row space of A.

(B) The column space of C is a subspace of the column space of A.

(C) The row space of C is a subspace of the row space of B.

(D) If the column vectors of A are linearly dependent，then the column vectors

    of C are linearly dependent.

(E) If the row vectos of A are linearly dependent，the the row vectors of C

    are linearly dependent.



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推 KAINTS:AB=C AE1E2....En=C B=E1E2...En 行消去 (D)True (E)False
→ KAINTS:提個想法 不太確定qq
推 justbelieve:ans:是BCD?
→ gskman:BCE
→ gskman:ABC都不一定是方陣
→ KAINTS:補一下 行消去式列保秩 我打錯了 是(d)false (e)true
→ KAINTS:G大是正解!!
→ gskman:不是方陣應該不能分成基本矩陣吧 還是我定義又搞錯了XD
→ KAINTS:我沒看到維度 sorry 請忽略我
推 john97611017:why有E?
推 pikachu123:這題是Leon習題出來的
→ pikachu123:(E)因為A行相依 存在x!=0使得xA=0
→ pikachu123:2邊同乘B 得xAB=0也就是xC=0 這樣就存在x!=0使得xC=0
→ pikachu123:所以C行相依
→ pikachu123:(D)A改B就會對
→ pikachu123:(E)用Rank來看也可以 A:mxn B:nxp C:mxp
→ pikachu123:因為A列相依所以rank(A)<m又Rank(C)<=Rank(A)<m
→ pikachu123:所以C列相依
→ pikachu123:  上面打裡打錯 改行相依
推 pikachu123:               列 (連打錯2次... 我大概太喜歡行空間)
→ sneak: m
→ sneak: xp https://daxiv.com