Terry Tung>試卷(2022/05/13)

# 主題課程_線性映射：判斷線性#107849

1.

Which of the following are linear cambinations of and ?
(A) (B) (C) (D) 2.

Suppoe that T: R2 → R2 is a linet tansformatian such that T and , = ?
(A) (B) (C) (D) 3.

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.

4.

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.

5.

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.

6.

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 .

7.

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) 【非選題】
8.

Let .Show that is linear combination of A and I2

【非選題】
9.

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

【非選題】
10.

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

【非選題】
11.

Let I' be a linear operator on R3 such that , , 【題組】

(a) Find .

【非選題】
12.【題組】

(b)Find 【非選題】
13.Let Y be the vector space of 2x2 mattices with real entries, and P3 the vector space of real polynomials of dogree 3 or less. Detine the linear transformation T: V → P3 by == 2a+(6-d)t-(a+c)x2+(a+6-c-d)x3
Find the rank and nullity of T.

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