(b) (6%) Suppose that exists. Find the matrix representation of corresponding to the ordered base s{x+1，x - 1} and for P₂ and R², respectively. 

(b) (5%) When the equation Ax = b is unsolvable, we may consider the so-called least squares problen to find a set of solutions, having the least squares error, fom solving a normal equation. Suppose that rank(A) =k ＜min(m, n) and let A = BC be a full rank decomposition of A . Use the known matrices B, C, and b to describe the unique projection vector p of b onto R(A) with the least || b-pll2.

(a) (4%) Denote the angle between f1 and f2 by θ. Find all possible values of a2such that θ = π/4.

(b) (6%) Now set a = 1. Find functions g1and g2 such that {g1, g2} is an orthonormal set that satisfies Span(g1) = Span(f1) and Span(g1,g2) =Span(f1, f2).

(A) (4%) Find all possible values of β such that, the inverse of L, does not exist.

(A) (15%) Let u(t) ≡ 0 and the initial conditions be x1(0)=x2(0)=.Find the solutions of the differential equations.

(B) (5%) Let initial conditions be x1(0) = x2(0) = c1(0) =, and u(t) be the unit step function. Does the solutions of the differential equations converge to constant values as time approaches infinity? Justify your answers.

15. (10%) Evaluate the following integral

1.1) According to the passage, what are the three factors that could be the reasons behind conficting research findings on the acquisition of grammatical morphemes? Explain your answer by summarizing the relevant paragraphs in your own words.

1.2) According to the passage, what are the three indications of a morphological congruency effect in the literature? Explain your answer by summarizing the relevant paragraphs in your own words. 

2.1) Imagine yourself as the researcher of this study. Based on the morphological congruency hypothesis, how do you predict the Russian and Japanese ESL speakers to perform on the self-paced reading task? Who will performn better and why? Also, in addition to the Japanese and Russian ESL speakers, is it necessary to include a group of English native speakers to test the hypothesis? Why or why not? Please be specific in explaining your answers.