課程名稱︰工程數學-線性代數 課程性質︰電機系大一必修 (統一教學) 課程教師︰蘇柏青、馮世邁、林茂昭 (三個班) 開課學院：電機資訊學院 開課系所︰電機工程學系 考試日期（年月日）︰2015/06/24 考試時限（分鐘）：110分鐘 試題 : 1. Let T:P2->P2 be a linear operator defined by T(a+bx+c^2) = (3a+c) + (a+b)x +(-a-b+3c)x^2 (x^2表示x平方) Let B1={1,x,x^2} and B2={1+x+x^2,1+2x+3x^2,2+x+x^2} be two bases for P2. (a)( 7%) Find TB1, the matrix representation of T with respect to B1. (b)( 8%) Find TB2, the matrix representation of T with respect to B2. 2.(10%) Please show two 3x3 matrices A and B respectively such that A is diagonalizable and B is not, where A and B have the same characteristic polynomial. 3.(20%) Determine whether the following statements are true or false. (No explanation is needed. Each correct answer gets 2% and each wrong answer gets 0%.) (a) Let P and Q be nxn orthogonal matrices, then P+Q is an orthogonal matrix. (b) The dimension of the vector space Mn*n is n^2. (c) The set of all symmetric matrices is a subspace of Mn*n. (d) T: Mn*n->Mn*n defined by T(A)=PA for A 屬於 Mn*n is a one-to-one linear transformation, where P is an orthogonal matrix. (e) Let Wi be a subspace of a vector space V for i=1,2. Let v be a vector in V. Let wi be the orthogonal projection of v on Wi for i=1,2. Then, w1+w2 be the orthogonal projection of v on Span{W1 聯集 W2}. (f) A diagonalizable matrix is similar to a diagonal matrix. (g) Let A be mxn and W=Col(A). Then Ax = Pw b is consistent for each b in R^m. (h) Let S be a nonempty subset of R^n. Then S 交集 S(perp) = {0向量}. (i) Let S = {v1,v2,...,vn} be a basis for R^n consisting eigenvectors of an nxn symmetric matrix. Then [v1 v2 ... vn] is orthogonal. (j) Let W be a subspace of R^n and v 屬於 R^n. Let w be the orthogonal projection of v onto W. Then ||w||≦||v||. 4. A matrix is said to be skew-symmetric if A^(T) = -A. Let W be the set of all 3x3 skew-symmetric matrices. Let T:W->W be defined by T(A)=CAC^(T), [ 1 1 2] where C = [ 2 1 1] . [ 1 0 -1] (a)( 8%) Find a basis B for W. (b)( 6%) Find [T]B, where B is the basis for W in Part(a). (c)( 6%) Find a basis for the null space of T. 5.( 5%) Apply the Gram-Schmidt process to replace the given linearly independent set S by an orthogonal set of nonzero vectors with the same span. [1] [0] [1] S = { [1] , [1] , [0] } [0] [1] [1] [1] [1] [1] 6. Let V be a vector space and T:V->V and U:V->V be linear operations. Prove or disprove the following statements. (a)( 5%) If u and v are both eigenvectors of T, then (u+v) is also an eigenvector of T. (b)( 5%) If v is an eigenvector of both T and U, then v is an eigenvector of T+U. (c)( 5%) If v is an eigenvector of both T and U, then v is an eigenvector of T。U. 7. Let C([-1,1]) = {f:[-1,1]-> R(實數)| f is continuous} be the set of all continuous real-valued functions defined on [-1,1]. We learned in class that it is a subspace of the function space F([-1,1]) with standard definitions of function addition and function scalar multiplication. Now, consider the following operations defined on two vector with a real-valued output. Determine each of them whether it is an inner prodect on C(R(實數)). If it is an inner product, show that all the four axioms are satisfied; if not, give an axiom that the operation violates explicity. 1 (a)( 5%)〈f,g〉= ʃ f(t)g(t)dt. -1 1 (b)( 5%)〈f,g〉= ʃ f(t)g(t)dt. -1 1 (c)( 5%)〈f,g〉= ʃ f(t)g(t)dt. -1 Hint: An operation 〈˙,˙〉is called an inner product on V if it satisfies (1)〈u,u〉>0 if u ≠ 0向量 (2)〈u,v〉=〈v,u〉 (3)〈u+v,w〉=〈u,w〉+〈v,w〉 (4)〈au,v〉= a〈u,v〉 for all u,v,w 屬於 V, a 屬於 R(實數). --

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