Terry Tung>試卷(2022/05/13)

# [無官方正解]主題課程_線性映射：one to one and onto#107850

1.

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.

【非選題】
2.
109成大電機_線性代數
(21 %) Let T: U→Vbe a linear transformation of vector spaces.

【題組】(a). Prove that if ker T= {0}; then T is one-to-one.(7 pts)

【非選題】
3.【題組】(b). Suppose T is one-to-one and {u1,... ,uk} is a linearly independent set of vectors in U. Prove that [T(u1),....,T(uk)] is a linearly independent set of vectors in V.(7 pts)

【非選題】
4.【題組】

(c). Define UP(t)(a polynomil of degree 2 and its standard form is P(t) = a0+ a1t + a2t2),V R3, and T(U) = Find U such that the image under T of Uis [11, 1, -1]T. (7 pts)

【非選題】
5.
106台聯大_工數A”B
(10%) Define a linear transformationT : R3→ R3by T((1,0,0)) =(0,1,0),T((, 1, 0)) - (0,0,1), and T((0,0, 1)) = (1,0,0).

【題組】

(a) (5%) Find a vector u = (ux, uy, uz) such that T(u) - u and.

【非選題】
6.【題組】(b) (5%) Is T : R3 → R3 one-to-one and onto? Why or why not?

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