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 function
f(x) ∈ C(R).

(E) None of the above.

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) Every function from , T : , has a standard matrix A such thatT(x) = Ax for all x ∈ .(B) The matrix transformation induced by an m X n matrix A (i.e., ) is a lineartransformation.(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 vectorsin its domain.(E) If f is a linear transformation and f(u) = f(v), then u = v.

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 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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