課程名稱︰線性代數一 課程性質︰數學系大一必修 課程教師︰莊武諺 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2018/11/7 考試時限（分鐘）：150 試題 : (以下的ε均代表屬於) (滿分115分) (1)(15 points) Let T: R^2→R^3 be the linear transformation defined by T((1,3))=(2,2,2) and T((3,1))=(1,-3,0). Letα={(-2,2),(2,2)}andβ={(2,3,1),(1,2,1),(1,1,1)} be ordered bases of R^2 and R^3 respectively. Find the matrix representation β [T] α (2)(15 points) Let T: R^5→R^3 be the linear transformation given by T(x)=Ax, 1 -2 3 2 1 where A=(4 2 3 -3 1) 2 0 0 2 1 Please find bases of the null sapce N(T) and the range R(T). (3)(15 points) Let V be a finite-dimensional vector space and T:V→V be a linear transformation. Suppose that rank(T)=rank(T^2). Prove that V=R(T)⊕N(T). (4)(15 points) (a) Let A εMnxn(F) satisfying A^7=0. Show that In-A is invertible and compute (In-A)^-1. (b) Let A,B εMnxn(F). Suppose that A+B is invertible. Show that A(A+B)^-1B=B(A+B)^-1A. (5)(15 points) Let A be an invertible upper-triangular nxn matrix. Show that A^-1 is also upper-triangular, (6)(20 points) Let A,B,C,D εMnxn(F) and OεMnxn(F) be the zero matrix. A O (a) Show that det(C D)=det(A)det(D). (b)Further assume that D is invertible and CD=DC. A B Show that det(C D)=det(AD-BC). (7)(20 points) Let V be an m-dimensional vector space over an infinite field F and{u1,...,un}be a linearly independent subset of V. Prove that for any v1,...,vnεV,{u1+αv1,...,un+αvn}is linearly independent over F for all but finitelymany values αεF. (Hint: You can only prove the statement for the m=n case to get partial points. Feel free to apply the fact that a degree d polynomial with coefficients in F has at most d zeros in F.) --

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