8. (5%) Which of the following is a linear operator?
(A) T : RR where T(x) = 4x + 3 for any .
(B) T : R² →R² wherefor any vector
(C) T :P → P where T(f(x)) = f(x)(x² + 1) for any polynomial
(D) T : where T(f(x)) = f'(x)+ f(x) for any differentiable function
(E) None of the above.

複選題1. (5%) Which of the following is true? (A) For any vector, {v} is linearly independent. (B) If a subset of R" contains more than n vectors, then it is linearly dependent. (C) If every column of an m x n matrix A contains a pivot position, then the matrix equation Ax = b is consistent for every. (D) A homogeneous equation (i.e, a system of linear equation Ax = 0) is always consistent. (E) A homogeneous equation always has infinitely many solutions.

2. (5%) Which of the following is true? (A) A matrix is invertible if and only if its reduced row echelon form is an identity matrix. (B) For any two n x n matrices A and B, if AB = In, then A is invertibie and = B. (C) If a square matrix has a column consisting of all zeros, then it is not invertible. (D) If A and B are invertible matrices, then A + B is also invertible. (E) If A is an n X n matrix such that Ax = b is consistent for every b in Rn, then Ax = b has a unique solution for every b in Rn.

複選題3. (5%) Which of the following is true? (A) Every function from Rnto Rm, T : Rn →Rm, has a standard matrix A such that T(x) = Ax for all . (B) The metrix transformation induced by an m x n matrix A is a linear traneformation. (C) The image of the zero vector under any linear transformation is the zero vector. (D) A function T : Rn -→ Rm 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.

複選題4. (5%) Which of the following is true? (A) A linear transformation with codomain Rm 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 : Rn → Rm and U : RP → Rq 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. (5%) Which of the following statements about the determinant is true? (A) For any square matrix A, det AT = -det A. (B) The determinant of the n✖n identity matrix is (-1)n. (C) The determinant of an upper triangular matrix is always equal to the product of its diagonal entries. (D) Performing a row addition operation on a square matrix does not change its determinant. (E) Performing a scaling operation on a square matrix does not change its determinant.

6.(5%) Which of the following is true? (A) Every nonzero subspace of Rn has a unique basis. (B) Every subspace of Rn has a basis composed of standard vectors. (C) The column space of an m ✖ n matrix is contained in Rm. (D) If V is a subspace of dimension k, then every generating set for V contains exactly h vectors. (E) The pivot columns of a  m ✖ n matrix A form a basis for the column space of A.

7. (5%) Which of the following is an eigenvect ctor of ?(A)(B)(C)(D)

9. (5%) Which of the following sets is an orthonormal basis for the designated vector space? (A) , for R2 with (x,y) -xTy. (B) {cosz,sin x},for C[0,2π] with (f,g) = f(x)g(x)dx. (C) {1,x,x2] for P3with(D) for R3 with (x,y)=xTMy where M = (E) None of the above.

複選題10. (5%) Which of the following is a vector space? (A) P = {p I p is a polynomial in z of any order} (B) Pn = {p | p is a polynomial in ； of exactly nth order} (C) {TIT : Rn → Rm,T is a linear transformation}(D) The set of all m x n n matrices of which the sum of all entries is zero, ie,(E) The set of all n x n square matrices with trace 1, ie,

20、Consider the following partial differential equation for the bivariate function v(x, t)subject to conditionsExpress the solution v(x, t) in the following formfor some constants an, bn, pn, qn, rn ∈ R satisfying 0 ≤ r0 ＜ r1 ＜ r2 ＜ ... and ＞0for n = 1, 2, ... Which of the following statements is/are true?(A) a0 = - (B) b1 = (C) p3 = (D) r4 = (E) None of the above is true.

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