複選題

19. Consider the linear equation Ax = b with A ∈ . Which of the following statements are true?
(A) If rank = m, then there exists at least one solution.
(B) Ifrank = n, then there exists exactly one solution.
(C) Ifrank = n, then the column vectors of A are linearly independent.
(D) Ifn ＞ m, then there exists at least one solution.

(E) Ifm ＞n, then there exists at most one solution.

15.考虑將前述方程式就=(0,0)線性化後之線性方程式。假設b=2,且該方程式之輸入項(forcing term)為sint。下列敘述何者為正確? (A)該線性方程式的解會收斂到cost。 (B)該線性方程式的解收斂到sin t。 (C)如該線性方程式的初值為(0,0),則方程式的解為sin t。 (D)如該線性方程式的初值為(0,1),則方程式的解為cos t。

複選題下面16-23題為複選題, 16. 令。下列關於之敘述何者正確? (A)該矩陣是一個3x3的方陣。 (B)該矩陣在t≥0時,永為一個可逆矩陣。 (C)該矩陣(3,2)位置那一項為 (D)該矩陣(1,2)位置那一項為。 (E)當t→∞時,該矩收斂為0矩陣。

17.考慮微分方程組。下列敘述何者正確?  (A)若(α,β)=(0,0),則有些「非零初值」對應之解會發散。(B)若(α,β)=(0,0)則任何非零初值」對應之解都不會發散。(C)若(α,β)=(1,1)·則方程式之解是頻率為之線性組合。(D)若(α,β)=(1,1),則方程式之解是頻率為sin√2t與cos√2t之線性組合。(E)若(α,β)=(-2,-2),則方程式之解是頻率為1與√3之旋波的線性組合。

複選題18. Consider the linear equation Ax = b, where A = [a1, a2, a3, a4 ] ∈ and a1, a2, a3, a4 are column vectors of A. Suppose a1 + a2 + a3 + a4 = b. Which of the following statements are true? (A) The linear equation has exactly one solution. (B) The linear equation has infinitely many solutions. (C) No conclusion can be drawn about the number of solutions to the linear equation. (D) The vectors a1, a2, a3, a4 are lincarly dependent. (E) rank([A,b]) =

複選題20. Consider the linear mapping L: V → W. Let be the zero vectors in V and W, respectively.Which of the following statements are true?(A) The condition L(V1) = L(V2) impliesV1 = V2.(B) For any W ∈ W, there exists V ∈ V such that L(V) = W.(C) If L is one-to-one, then L(v) = 0w implies v = Ov.(D) If V1,V2,..., are linearly independent, L(V1), L(V2),.., are also linearly independent.(E) The condition C1V1+C2V2+..+=implies c1L(V1)+c2L(V2)⋯+=.

複選題21. Given vectors x, y,z in and matrices A, B, C in . Which of the following statements aretrue?(A) (B) (C) (A+B)(A-B) = A2 - B2(D) If AC = BC and C is not the zero matrix, then A = B.(E) If AB equals the zero matrix, then BA also equals the zero matrix.

複選題22.Let A ∈ , denote the column space of A, denote the null space of A,and dim(S) denote the dimension of a subspace S. Which of the following statements are true?(A) For any x ∈ , there exists u ∈ and v ∈ such that x = u + v.(B) Suppose u ∈ and v ∈ . Then = 0.(C) dim+ dim =n(D) For any y ∈ , there exists x ∈ such that y = .(E) Ify ∈ , then y is the zero vector in .

複選題23. Let A ∈ and x ∈ . Which of the following statements are true?(A) If A is singular, then O is an eigenvalue of A.(B) A and share the same eigenvalues and eigenvectors.(C) If A is diagonalizable, then A has n distinct eigenvalues.(D) Suppose that A is nonsingular. The condition Ax = Ix implies .(E) Supposc that all the eigenvalues of A are real and positive. Then we have ＞ 0.

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