複選題

 Which of the following is true?
(A) Every function from , T : , has a standard matrix A such that
T(x) = Ax for all x ∈ .
(B) The matrix transformation induced by an m X n matrix A (i.e., ) is a linear
transformation.
(C) The image of the zero vector under any linear transformation is the zero vector.
(D) A function T : is uniquely determined by the images of the standard vectors
in its domain.

(E) If f is a linear transformation and f(u) = f(v), then u = v.

複選題108台大電機資訊學院_工數C .(10%) For the matrix below, which of the following are correct?(A)-1 is its eigenvalue. (B) 1 is its eigenvalue. (C)-3 is its eigenvalue. (D)Zero vector belongs to its eigenvector. (E)[1 0 2]T is a basis of its eigenspace.

複選題108台大電機資訊學院_工數C (5%) Which of the following are correct? (A) An eigenvalue of a matrix has infinite number of eigenvectors. (B) If det(A -λln) = O, λ is the eigenvalue of n x n matrix A. (C) IfA is the cigenvalue of matrix A, the columns of (A - λIn) are independent. (D) Ifa is the eigenvalue of matrix A, there exist a non-zero vector V such that Av = av. (E) The number of the eigenvalue of n x n matrix A may larger than n.

複選題Which of the following are linear cambinations of and? (A)(B)(C)(D)

Suppoe that T: R2 → R2 is a linet tansformatian such that T and , = ?(A)(B)(C)(D)

複選題Which of the following is true?(A) A linear transformation with codomain is onto if and only if the rank of its standardmatrix is m.(B) A linear transformation is one-to-one if and only if its null space consists only of the zerovector.(C) A linear transformation is onto if and only if the columns of its standard matrix form agenerating set for its range.(D) If the composition UT of two linear transformations T : and U : isdefined, then m = p must be true and the composition UT is also a linear transformation.(E) For every invertible linear transformation T, the function is also a linear transfor-mation.

複選題Which of the following is a linear operator?(A) T : R→R where T(x) = 40 + 3 for any I E R.(B) T:T2→R2 where for any vector x∈ R2?.(C) T : P →P where T(f(x)) = f(x)(x2 + 1) for any polynomial f(x) ∈ P.(D) T : where T(f(x)) = f'(x)+ f(x) for any differentiable functionf(x) ∈ C(R).(E) None of the above.

複選題Which of the following are correct?(A) The system defined by F(x,y)=(x2, x) is linear.(B) The system defined by F(x,y)=(dx/dt, x) is linear.(C) A and B are m x n matrices. If Aw = Bw for all w in , then A = B.(D) The columns of matrix A contains zero vector. If Ax=b have solution, it will have Infinite solutions.(E) The zero vector of R" is within the span of any finite subset of .

複選題Which of the following transformations are not linear?(A) S: the map in R3 which rotates points about the x1-axis by an angle π/2.(B) (C) (D) T(ax2+bx+c)-(a+b)x+(b+c)(E)

複選題109中山電機Consider the lincar mapping L: V→W. Let Ov and Ow be the zero vectors in V and W, respectively. Which of the following statements are true? (A) The condition L(v1) = L(v2) implies v = V2. (B) For any w W, there exists v  V such that L(v) = w. (C) If L is onc-to-one, then L(v) = 0w implies v = 0v. (D) Ifv1,v2, , Vk are linearly independent, L(v1), L(v2), ...,L(vk) are also linearly independent. (E)The condition c1v1+c2v2+ckvk=0vimplies c1L(v1)+c2L(v2)+⋯+ckL(vk)=0w.

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