課程名稱︰線性代數 課程性質︰電機系必修 課程教師︰馮世邁 開課學院：電機資訊學院 開課系所︰電機系 考試日期（年月日）︰6/19 考試時限（分鐘）：100 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.Prove that a linear operation on a finite-dimensional vector space is invertible if and only if zero is not an eigenvector of T. (7%) 2.Let ┌ ┐ A=│2 1│ │1 2│ └ ┘ Let B=A^100. Please find B11, the entry at the first row and the first colume of B.(8%) 3.The set of all n*n matrix Mn*n with the conventional matrix addition and scalar-matrix multiplication os a vector space. Please answer whether the follwoing statements are true or not and also justify your answers. (a) The set S{A:A=A(t),A屬於Mn*n}under the same matrix addition and scalar mutiplication is a subspace of Mn*n. (5%) (b) The set Q{A:A is invertible,A屬於Mn*n}under the same matrix addition and scalar multiplication is a subspace of Mn*n. (5%) 4.Let L(Mn*n, Mn*n) be the set of all linear transformation from Mn*n to Mn*n. Let T,U屬於L and c be some scalar. Define T+U: Mn*n → Mn*n and cT: Mn*n → Mn*n by (T+U)(A)=T(A)+U(A) (cT)=cT(A) for all A屬於Mn*n Thus, V=L is a vector space over R. What is the dimension of V? No proof needed. (5%) 5.Let T be a linear operator on an inner product space V. Suppose that ||T(x)||=||x|| for any x屬於V. (a) Prove that T is one to one. (b) Prove that <T(x),T(y)>=<x,y>for any x, y in V. (c) Let B={b1,b2.....bn} be an orthogonal basis of V. Prove that the matrix representation of T with respect to B is orthogonal. (7%) ┌ ┐ 6.Find U(│X1│), where U is the orthogonal projection of R^2 on the line with │X2│ └ ┘ equation y=-3x.(10%) ┌ ┐ ┌ ┐ │10│ │3│ 7.Find z, such that ||Az-b|| is a minimum, where A=│01│, b=│3│ (10%) │11│ │3│ └ ┘ └ ┘ 8.Consider the vector space of 2*2 real matrices. The inner product of A,B is defined as <A,B>=a11b11+a12b12+a21b21+a11b22. Find the 2*2 real symmetry matrix that is closest to ┌ ┐ │12│ │48│ └ ┘. (10%) 9.(a) If λ1 is one eigenvalue of a invertible real matrix A, the corresponding eigenvector is P1 show that 1/λ1 is one eigenvalue of A^-1, the corresponding eigenvector is P1.(10%) (b) If A is a diagonalizable matrix, show that A^k is diagnalizable for any positive integer k. (10%) --

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