無年度 - [非官方正解]主題課程_線性映射：基底變換和座標變換#107933

科目：主題課程專用 | 選擇題數：6 | 申論題數：10

試卷資訊

所屬科目：主題課程專用

選擇題 (6)

A is a linear-transformation matrix in the standard coordinate system. 
If the coordinate system is changed to βcoordinate system,
 to find the transformation matrix (called the  β-matrix) relative to  β for 
the transformation x |-Ax, where β= {b1, b2, b3},

31.0, -1 are in the  β-matrix. (A) True. (B) False.
32.1, -3 are in the  β-matrix. (A) True. (B) False.
33. -2, -4 are in the  β-matrix. (A) True. (B) False.
34.-6, -7 are in the β-matrix. (A) True. (B) False.
35. 8, -9 are in the  β-matrix. (A) True. (B) False.
複選題

,Let A = and define a transformation T : R2 → R3 by T(x) = Ax. Which 1-1 of the following statements is/are true? (A) The image of x = under T is . (B) There is exactly one x whose image under T is. (C)The vector b is in the range of T if b is the image of some x in R2. (D) The vector b is in the range of T if the system Ax = b is inconsistent. (E) The vector is not in the range of T. 

申論題 (10)

(a) (4%) The matrix representation of T' relative to ordered bases B, C.
(b) (4%) The matrix representation of T relative to ordered bases B, D.
(a) (3%) Find a basis for V.
(b) (12%) Find the eigenvalues of T. 
(a) If A and B are n x n matrices that have the same rank, then the rank of A? must equal the rank of B2.
(b) Let L : R2→ R2 be a linear operator, and let A be the standard matrix representation of L. If L2 is defned by
 L2(x) = L(L(x)) for all x62847eb6ece2b.jpg R2
  then L2 is a linear operator and its standard matrix representation is A2.
(c) If L1 and L2 are both linear operators on a vector space V, then Li + L2 is also a linear operator on V, where L1 + L2 is the mapping defined by 
(L1+L2)(v)=L1(v)+L2(v)forallv62847eb6ece2b.jpgV
(d) If N(A) = {0}, then the system Ax = b will have a unique least squares solution.
(e) If{u1, u2,... ,uk} is an orthonormal set of vectors inRnand
 U = {u1, u2,... ,uk} 
 then UUT = In (the n x n identity matrix).

(15%) Let T:P2(R)＞ P2(R) be a linear operator that is defined according to T(g(x))=g(x)+628480efe1df4.jpg , in which P2(R) is a set of all polynomials with real-value coefficients and degree n, n=0,1,2. Please find the eigenvalues and the associated eigenvectors of the operator6284810ac472d.jpg