107年 - 107 國立中山大學_碩士班招生考試_電機系(乙組)：工程數學乙#113265

(b) (2%)Use the Fundamental Subspaces Theorem to show that N(A^TA) N(A).

(a) (1+2%) Let δ be a unit element of"t", the addition operation of V. (i) Write the equality condition about & as shown in the corresponding axiom. (ii) Let y be another unit element of "+". Show that δ=y.(須註明使用到的所有向量空間定義中的公設,否則不計分)

(c) (2%) Let 〈●，●) be an inner product defined on V. Show that the function defined satisfies the triangular inequality property.

(d) (2+3%) Consider the vector space R^2✖2 with the inner product (A,B):= trace(A^TB) and denote Y:= .(i) Describe Y^⊥ as the span of an orthonormal basis. (ii) What is the matrix, denoted by P_Y⊥, that represents the orthogonal projecting operation Ⅱ_Y⊥ : R^2✖2 → Y^⊥ along with the subspace Y with respect to the ordered basis E = {G,H}, where G and H are vectors of the standard bases for Y and Y^⊥, respectively?

(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 {μ_i，....，μ_k}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. 

(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 . 

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

(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).