無年度 - 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)

Let .Show that is linear combination of A and I₂. 

Suppose that T: R²→R³ is a linear transformation such that and Determine for any in R². 

Let T be a linear operator on R³ such that T, Please find the standard matrix of T.

(a) Find .

(b)Find