[試題] 101下 馮世邁 線性代數期末考
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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