31. Consider {0} and {1} as the basis for a vector space in C², then the natural basis for the tensor product space C² ⊗ C² has the ordered bases {00}, {10}, {01}, {11}. The tensor product of two vectors is defined from their decomposition on the bases.
What is the corresponding matrix formed by

wheredenotes the transpose.


(A)

(B)

(C)

(D)

(E)

32. Let R⁴ˣ⁴ denote the collection of all 4 × 4 real matrices. Define the lineartransformation T: R⁴ˣ⁴ → R⁴ˣ⁴ by T( A ) = Aᵀ + 2A for any 4 × 4 matrix A.What is the rank of T?(A) 4(B) 8(C) 10(D) 12(E) 16

33. Let R⁶ denote the 6-dimensional real vector space and ||•|| denote the standard vector norm in R⁶. How many 6 × 6 real matrices U satisfy || Ux || = || x || for all x ∈ R⁶ and |trace(U)| = 6?(A) 1(B) 2(C) 3(D) 6(E) Infinitely many

34. Let Rθ denote the rotation matrix through an angle 0＜θ＜counterclockwise about the origin in R². Let e₁ = [1,0], e₂ = [0,1]ᵀ be the standard basis of R². Suppose A is a 2x2 matrix satisfying Ae₁ = Rθe₁ and RθAe₂ = e₂. The followings are 5 statements about A that may or may not be true.(1) A²e₁ = R₂θe₁(2) det( A ) = 1(3) trace( A ) = 2cos θ(4) A is a positive definite matrix(5) ARθ = RθAWhich of the following combination of statements about A are all true?(A) (1)(2)(B) (1)(3)(C) (2)(3)(D) (3)(4)(E) (4)(5)

35. Given a matrix Find the eigenva lue of A².(A) 1 and 2(B) -1 and -2(C) 2 and 4(D) 1 and 8(E) 1 and 16

36. What are the dimensions of the row space, column space, and null space for the matrix?(A) 3, 5, 5(B) 3, 5, 2(C) 3, 3, 2(D) 2, 3, 3(E) 2, 2, 3

37. A linear equation system Ax = b consists of m linear equations with n variables. Which of the following statements is correct?(A) We can apply elementary column operations to solve this linear equation system.(B) If A and B matrices are similar, then Ax=b and Bx=b have the same solution set.(C) If this linear equation system has solutions, the solutions form a subspace.(D) If m ≠ n and rank( A ) = m, then there exist infinite solutions.(E) If m ≠ n, we can use "Cramer's rule" to find its solution.

38. If a linear transformation T: R³→R³ is defined by:T(x, y, z) = (3x+2y, -2x+3y, 5z), which of the following statement is incorrect?(A) The basis of the kernel (null space) of T is {0}.(B) T is one to one.(C) T is onto.(D) T is invertible.(E) T is not diagonalizable over the real field.

39. Which of the following statements is true?(A) Let S = {v₁, v₂, ..., vk} be a set of vectors in Rⁿ. Let A ∈ Mmxn(R). If S is linearly independent, then S' = {Av₁, Av₂, ..., Avk} is also linearly independent(B) Let S be a set of vectors in a finite-dimensional vector space V. Then, span(S) is the intersection of all subspaces of V that contains S(C) Assume that S = {v₁, v₂, ..., vn} is a subset of a vector space V. If S is linearly dependent, then there exists a proper subset of S that is a basis for V(D) The basis of any vector space is unique(E) Assume that S = {v₁, v₂, ..., vn} is a subset of a vector space V. If S is linearly independent, then there exists a proper subset of S that is a basis for V

40. Consider the continuous function f(x) = 3 cos 2x + 4 sin 2x, and let g(x) = a cos x + b sin x be a linear combination of the functions cos x and sin x.Define inner product of two functions as (f, g) = f(x)g(x)dx. Please find a and b such that ||f - g||² = |f(x) – g(x)|² dx is minimized. Which of the following shows the correct answer?(A) a = 4 and b = 3(B) a = 3 and b = 4(C) a = 0 and b = 0(D) a = 3 and b = -4(E) a = -3 and b = 4

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