主題課程_線性映射：判斷線性#107849　

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)

 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.

Which of the following is true?

(A) A linear transformation with codomain is onto if and only if the rank of its standard
matrix is m.

(B) A linear transformation is one-to-one if and only if its null space consists only of the zero
vector.

(C) A linear transformation is onto if and only if the columns of its standard matrix form a
generating set for its range.

(D) If the composition UT of two linear transformations T : and U : is
defined, 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.

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.

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 .

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)