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

(a) (2%) (i) What is the mathematic relationship of dimensions of two subspaces R(A) and N(A) for any A matrix considered here? (ii) What is the mathematic relationship described in the Fundamental Subspaces Theorem about matrix A ?

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

(c) (2%) (i) What is the condition on R(A) or N(A) that is equivalent to the existence of solution to Ax= b? (ii) What is the condition on R(A) or N(A) that is equivalent to the uniqueness of solution to Ax = b, if it is solvable?

(d) (2+4%) When the equation Ax = b is unsolvable, we may consider the so-called least squares problem to find a set of solutions, having the least squares error, from solving a normal equation. (i) Use the property and condition mentioned in (b)-(c) to explain why the normal equation is always solvable.

(ii) Suppose that rank(A) = k ＜ min(m, n) and let A = BC be a full rank decomposition of A . Use the known matrices B, C, and b to describe the unique projection vector p of b onto R(A) with the least∥b-p∥₂.

(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)

(b) (3%) Let X and Y be two subspaces of V. (i) Write the definition of XtY . (ii) Write the definition of X⊕Y . (ii) What is the mathematical relationship between dim (X⊕Y) and dim(X⊕Y)?

(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) (1+2%) (i) Write the definition of superposition principle and (ii) show the implication from the combination of additive and homogeneous properties to the superposition-principle.

(b) (2%) Write (i) the definition of Ker(L), the kernel of L, and (ii) the definition of L(V), the image of vector space Y, respectively. Ib-plz

(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,...,V_n] and F =[w₁,. ,w_m] 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 {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.

(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).

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

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