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- 1.
(1) For an n× n upper triangular matrix A , prove that det (A) equals the product of
the diagonal elements of A.
(2) Prove that if V is a vector space of dimension n , then any set of n linearly
independent vectors spans V.
- 題型：申論題
- 難易度：尚未記錄
- 2.
For the following matrix, find a basis for the row space and nullspace.
- 題型：申論題
- 難易度：尚未記錄
- 3.
(1) For the system Ax = b , find the least squares solution, where
(2) Find the eigenvalues and corresponding eigenvectors of the following matrix:
(3) Give the definition of similar matrix, and show that similar matrices have the same
eigenvalues.
- 題型：申論題
- 難易度：尚未記錄
- 4.
If F(s) is the Laplace transform of f(t) , denoted by F(s) =L{ f(t)} , find the
inverse Laplace transform L−1{F(as + b)} in terms of f(t) , where a > 0 and
b ≠ 0 .
- 題型：申論題
- 難易度：尚未記錄
- 5.
Solve
- 題型：申論題
- 難易度：尚未記錄
- 6.
Let
Compute eAt .
- 題型：申論題
- 難易度：尚未記錄
- 7.
Consider the non-homogeneous linear system x´= Ax + eαtv , where x is a vector
consisting of functions in t , α is not an eigenvalue of A , and v ≠ 0 is a constant
vector. Find a particular solution of the system, in terms of A,α,v and t .
- 題型：申論題
- 難易度：尚未記錄
- 8.
(1) Determine the Fourier series coefficients (an , bn ) of the function f(t) = t•u(t)
expanded over the interval (−π,2π ) , where u(t) is the unit-step function.
(2) If the coefficients (an , bn ) from (1) are also the Fourier series coefficients of
some function expanded over the interval (−2π,4π ) , find the function in terms of
f(t) .
- 題型：申論題
- 難易度：尚未記錄
- 9.
Given y1(x) = xr is one solution of the homogeneous 2nd order linear differential
equation x2y´´− 5xy + 9y = 0 .
(1) Derive its characteristic equation in terms of parameter r.
(2) Let y2(x)=v(x)y1(x) be another linearly independent solution. Determine the
governing differential equation of v(x) .
(3) Find v(x) by solving the differential equation in (2).
(4) Apply the method of variation of parameters to find a particular solution of
- 題型：申論題
- 難易度：尚未記錄
- 10.
(1) Find the recurrence relation of cn .
(2) Find the two linearly independent solutions. Please write the first three nonzero
terms if it is an infinite series.
(3) Find the guaranteed radius of convergence.
- 題型：申論題
- 難易度：尚未記錄