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102年 - 102 國立中山大學_碩士班招生考試_電機系(乙組)：工程數學乙#110521
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題組內容
7. (5%) Prove the following statements
(a) (3%) The Laplace transform is a linear operation.
其他申論題
(d) (6%) Let q(x) be the vector in S⊥ that is closest to . Compute||q(x)||2.
#473379
(a) (5%) Suppose u O and the equations are driven by non-zero initial conditions. Determine the conditions on the coefficients such that =0.
#473380
(b) (10%) Let the initial conditions be equal to zero. For the values1 = 0, = -1, and calculate the response y(t) = 2x1(t) - x2(t). Determine at what time the peak value of y occurs.
#473381
(c) (5%) For the values = -2, and u(t) = sin(t), calculate the steady-state response of y(t) =x1(t) - x2(t).
#473382
(b) (2%) Suppose the Laplace transform of a function y(t) is equal to Y(s). Then the Laplace transform of y(t - a) is equal to .
#473384
(a) (5%) Suppose that m=n = 3, A = . Find the set of all solutions to the -4 2 equation Ax = b if it is consistent. Otherwise, find vector p to solve and, moreover peR(A) compute the value of .
#473385
(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.
#473386
(a) (4%) Denote the angle between f1 and f2 by θ. Find all possible values of a2such that θ = π/4.
#473387
(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).
#473388
(A) (4%) Find all possible values of β such that, the inverse of L, does not exist.
#473389
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