無年度 - [非官方正解]主題課程_線性映射：one to one and onto#107850

科目：主題課程專用 | 選擇題數：1 | 申論題數：5

試卷資訊

所屬科目：主題課程專用

選擇題 (1)

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.

申論題 (5)

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

(c). Define U627dd5e300d8e.jpgP(t)(a polynomil of degree 2 and its standard form is P(t) = a0+ a1t + a2t2),V 627dd653e4c8c.jpgR3, and T(U) = 627dd6a394b32.jpg Find U such that the image under T of Uis [11, 1, -1]T. (7 pts) 

(a) (5%) Find a vector u = (ux, uy, uz) such that T(u) - u and627de8e3bb42b.jpg.
(b) (5%) Is T : R3 → R3 one-to-one and onto? Why or why not?