(a) (15%) Let u(t) = O and the initial conditions bex₁(0)=x₂(0)=1,.Find the solutions of the differential equations.

(c) (1+2%) Suppose L is linear.(i) Use Ker(L) to describe a mathematic condition that is equivalent to L being one-to-one. (ii) Moreover, show the implication from the condition on Ker(I) to L being one- to-one.

(d) (2+4%) Let L be linear with A as its matrix representation with respective to bases E = [V1,...,Vn] and F =[w1,. ,wm] for V and W, respectively. (i) Describe a mathematic condition about A that is equivalent to L being onto. (ii) Moreover, show the implication from that condition to L being onto.

(a) (2%) Let be an eigenvalue of A with corresponding eigenvector x. Derive in details the equation for solvingλ.(要求推導出公式、而非背寫出公式卻沒有任何數學的說明)

(b) (2+2%)(i) Let μ be an eigenvalue of matrix A . What is the mathematic notation for describing the number of linearly independent eigenvectors associated with μ? (i) Let {μi，....，μk}be the set of all distinct eigenvalues of matrix A. Use the corresponding notation as in (i) to describe the condition for A being a diagonalizable matrix. 

(b) (5%) Let initial conditions be x1(0)=x2(0)==0, and u(t) be the unit step function. Does the solutions of the differential equations converge to constant values as time approaches infinity? Justify your answers.

(a) (10%) Express the differential equations using the polar coordinate； i.e. express the equations in terms of (r, θ), where x1(t) =r(t) cos(θ(t)) and x2(t) =r(t) sin(θ(t).

(b) (10%) Solve the differential equations for any nonzero initial conditions (x10, x20).

(c) (10%) Draw the phase plane portraits for the solutions (x1,x2) of the following initial conditions:(x10,x20)=(0,0.5),(-1,0), and (1,-1).

(c) (5%) Suppose A = B+iC is a Hermitian matrix and let . Show that any eigenvalue λ of Ω is also an eigenvalue of A . 

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