(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.

(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 R2✖2 with the inner product (A,B):= trace(ATB) and denote Y:= .(i) Describe Y⊥ as the span of an orthonormal basis. (ii) What is the matrix, denoted by PY⊥, that represents the orthogonal projecting operation ⅡY⊥ : R2✖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?

(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,...,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. 

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