109年 - 台聯大_工程數學D#110754

(B). (2%) If Col(A)-Col(B), the reduced row echelon form of C is [In|In].

(E).(2%) Any vector v in Rⁿcan be expressed as the sum of the orthogonal projection of v onto each column of.

(A). (5%) Consider a system of linear differential equations in the following vector form:
Here v(t) is an n-dimensional column vector and M is an n x n matrix. Assume that the matrix M can be diagonalized by a similarity transformation:

Here S is the transformation matrix and D is a diagonal matrix with λ₁,λ₂,... ,λn as its diagonal elements. The initial values of the equations form the vector v(0) and P(t) is a diagonal matrix with exp (λ₁t),exp(λ₂t), ,exp (λ_nt) as its diagonal elements. Please express the solution v(t) for t＞ 0 in terms of v(0), S and P(t) and explain why.

(B). (5%) Continued from (A), if the matrix M is given by
write down a transformation matrix S that can diagonalize M into D and also write down the corresponding diagonal matrix D.

(C). (5%) Consider a system of differential equations in the following vector form: Here the additional term f(t) is a known n-dimensional t-dependent column vector and the other symbols are the same as in (A). Please express the solution v(t) for t ＞ O in terms ofv(0), S, P(t), f(t) and explain why.

(A) (5%) can be expanded into a sine series as follows:Please determine the solution y(x) in the range of 0 ≤ x≤1 interms of C_m , m=1,2,...∞.