[理工] 線代
(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
10/20 17:50, , 1F
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※ 編輯: KAINTS 來自: 42.64.225.213 (10/20 18:50)
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※ 編輯: KAINTS 來自: 42.64.225.213 (10/20 19:00)
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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?
→
10/20 21:47, , 11F
10/20 21:47, 11F
※ 編輯: KAINTS 來自: 123.193.7.20 (10/20 21:50)
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10/20 23:25, , 15F
10/20 23:25, 15F
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