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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