無年度 - 主題課程_線性映射：基底變換和座標變換#107933

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, b_3},

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→ R³by 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 x

R²then L² is a linear operator and its standard matrix representation is A².

(c) If L₁ and L₂are 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)forallv

(d) If N(A) = {0}, then the system Ax = b will have a unique least squares solution.

(e) If{u₁, u₂,... ,u_k} is an orthonormal set of vectors inRⁿand

U = {u₁, u₂,... ,u_k}
then UU^T = I_n (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)+ , 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 operator