首頁 > 線上測驗 > 100年研究所工程數學歷屆試題(工數1)電機所.電子所. 電信所.光電所.通訊所(二年期) > 100年國立中興大學電機所乙丙丁組工程數學
(1) Complete the matrix A (fill in the two blank entries) so that A has
eigenvectors x1 = (3,1) and x2 = (2,1) :
(2) Find a different matrix B with those same eigenvectors x1 and x2 , and with
eigenvalues λ1 = 1 and λ2 = 0 . What is B10 ?
(1) Given two arbitrary finite numbers f0 and f1 , if fk = fk−1 + fk−2
(2) Suppose B∈ Rm×n . Show that BT B is positive definite if and only if B has
independent columns.
Given a matrix
Find
(1) The adjoint of A .
(2) Find the characteristic polynomial of A .
(3) Find the eigenvalues of A .
(4) Based on the result in 3., is A invertible? If yes, find the inverse of A .
(5) Compute A6 + 2A5 + A4 .
Let Pn be the space of polynomials of degree at most n . Let the linear map
L : P2 → P1 be defined as L( p(x)) = ( p(x) − p(0)) / x . Find the matrix representation
of L with respect to the bases {1+ x, x + x2 , 1− x2} for P2 and {x, 1− x} for P1 .
Find the general solution of the following differential equations:
(1) y´´+ 4y´+ 4y = 3xe−2x .
(2) ( y2 − x2 )dx − 2xydy = 0 .
(3) x2y´´+ xy´+ 4y = sin(2ln(x)) .
Solve the following system of differential equations on (−∞,∞)
(1) Find the general solution for the homogeneous system.
(2) Verify that the two solution vectors are linearly independent on (−∞,∞).
(3) Find a particular solution vector for the nonhomogeneous system.
(4) Find the general solution for the nonhomogeneous system.
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