[理工] 線代

[KAINTS (RUKAWA)]

(1)Any system of n linear equations in n unknowns has at least
   one solution.
  F:改成at most 就對了嗎?

(2)For a linear system of three equations and four unknown
   variables,we should obtain mutiple solutions.
  F:有可能無解?

(3)If v1,v2...and vp are vectors in a nonzero finite- dimensional
   vector space V,and S={v1,...vp}. If S is a linear independent
   set,then S is a basis for V.
   F:還要在加這個條件dim(S)=dim(V)?

(4)If A is an orthogonal matrix,then A is symmetric.
   F:要real才成立?

(5)Let A be an n by n matrix with characteristic polynomial
   f(t). Let g(t) be a polynomiql for which 0<deg(g(t))<deg(f(t)).
   Then, g(A)=/=0.
   F:這題在考什麼?

(6)Consider a n*nreal-valued matrix A. Wich of the following
   statements are equivalent to "A is nonsingular"
   (a)The system of n linear equations in n unknowns Ax=e1
   has a unique solution,where e1=(1,0,...,0)^T.
   (b)A^2+3A+I is nonsingular.
   (c)A^2+4A
    Ans:a
   (c)我用det(A^2+4A)=det(A(A+4I)=det(A)det(A+4I)=\=0
      =>det(A)=/=0,det(A+4I)=\=0 => A is nonsingular.
      這樣c不也對嗎?
    還有a,b要怎判斷?

(7)If S generates the vector space V,then every vector in V can
   be written as a linear combination of elements of S in only
   one way.
   T:是因為題目說"S generates the vector space V",才得到span(S)
     是V的一組基底嗎?
問題有點多，感謝回答!

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◆ From: 42.64.225.213
※ 編輯: KAINTS          來自: 42.64.225.213        (10/20 17:36)
推 bouwhat:1.改成at most也不對因為rank沒給
→ bouwhat:2.有可能無解
推 bouwhat:3.同你說的
→ bouwhat:4.體是R才對
※ 編輯: KAINTS          來自: 42.64.225.213        (10/20 18:50)
→ KAINTS:thx
※ 編輯: KAINTS          來自: 42.64.225.213        (10/20 19:00)
推 ddczx:4.正交跟對稱無關= =....ex[0 0 1,1 0 0,0 1 0]
→ Murasaki0110:這是矩陣所以有關喔 (A^T)A=I
→ KAINTS:D大 正交矩陣應該有關吧 像我們做正交對角化
→ KAINTS:若A為實對稱矩陣,則A可正交對角化,存在一P為正交矩陣,使得
→ KAINTS:P^TAP=D,其中D為對角矩陣
剛剛沒補完
A=PDP^T
A^T=(PDP^T)^T=(P^T)^TD^TP^T=PDP^T=A
所以A為對稱矩陣

所以正交跟對稱矩陣有關係吧?
→ Murasaki0110:我又搞錯了 鬱卒= =
※ 編輯: KAINTS          來自: 123.193.7.20         (10/20 21:50)
推 ddczx:實對稱<->可正交對角化,但正交矩陣不一定對稱,我上面有反例
→ KAINTS:喔喔
→ KAINTS:阿 我知道我的盲點是什麼了 謝謝d大
→ cha122977:1給F是因為可能無解(例：x+y=2; x+y=3)