100年國立交通大學電機所工程數學(微分線性) 線上測驗

1.: Let x(t) be a function of t satisfying the following differential equation
tx´´(t) − 2x´(t) + tx(t) = 0
(1) Assuming x(0) = 1, find the Laplace transform X (s) of x(t).
(2) Continued from part (1), find x(t) through the inverse Laplace transform of X(s) .

2.: Find the generals solution of the following nonhomogeneous linear differential
equation
x´´(t) + x(t) = 2sec(t) .

3.: Let x₁(t) and x₂(t) be functions of t satisfying the following system of
differential equations

Assuming x₁(0)=x₂(0)=1and x₁´(0) = x₂´(0) = 0 , find the solutions x₁(t) and
x₂(t) .

4.: Solve the following initial value problem

for some t₀ > 0

5.: Consider the linear equations below:

(1) Which equations are consistent? Explain. (A linear equation is consistent if it has a
solution.)
(2) For each equation that is consistent, find a solution that has the least (Euclidean)
norm.
(3) Among the equations that are not consistent, determine the ones that have a unique
least squares solution? Explain.
(4) For each equation you found in (3), compute the least squares solution.

6.

7.: Suppose A is a 3×3 real matrix, and satisfies
(A₂ + A+ I )(A+ 2I ) = 0 (e)
where I is the 3×3 identity matrix.
(1) Show that A is nonsingular.
(2) Express A⁻¹ as a polynomial of A . Is your expression unique?
(3) Let V be the set of all 3×3 real matrices satisfying (e). Is V a vector space?
Explain.
(4) Assume that the eigenvalues of A are distinct, find det(A⁻¹) , the determinant of
A⁻¹ .

8.: (1) The 4 vectors

9.: Are the following statements True (T) or False (F)? For each of your answers give a
brief explanation. A correct answer with no justification will receive no credits.
(1) If A and B are similar matrices, then det(A) = det(B) .
(2) If two matrices A and B have the same null space, then they must be either
both singular or both nonsingular.
(3) If y ≠ 0 is such that A^T y = 0 , then A_x = y has no solution.

購買題庫後，可使用那些功能？

可觀看題目詳解，並提供模擬測驗！（免費會員無法觀看研究所試題解答）