课程名称︰线性代数 课程性质︰必修 课程教师︰冯蚁刚(电机系四个老师考题都一样) 开课学院：电资学院 开课系所︰电机系一年级 考试日期（年月日）︰2009 6/17 考试时限（分钟）：100分钟 是否需发放奖励金：是 试题 : Linear Algebra Final Examination Dept. of Elec. Eng., National Taiwan University June 17, 2009 USE OF ALL AUTOMATIC COMPUTING MACHINES IS PROHIBITED 1.Suppose the real nxn matrix A is invertible and diagonalizable, where n>1. Briefly explain why the following matrices are all diagonalizable: A^5+8A^2, A^(-1), A^T, (A^T)A,and A^~ obtained by 1) exchanging the first and last rows of A and 2) exchanging the first and last columns of the resultant matrix. (25%) Four of these five matrices are always invertible and the other one may become not invertible. Identify the one and explain why. (5%) 2.In the real space R^n, define the inner product<x,y>=x1y1+...+xnyx and the norm ||x||=<x,x>^(1/2) for all vectors x=[x1 ... xn]^T and y=[y1 ... yn]^T. Suppose S is a subspace of R^n. Let S^⊥ be the orthogonal complement of S in R^n with respect to <‧,‧>. For any matrix A, let Null(A) denote its null space. Consider any real 4x4 matrix A with Null(A) spanned by the set {[2 0 2 -1]^T, [1 2 0 -1]^T,[3 -1 4 -1]^T}. (a)Find an orthonormal basis for Null(A)^⊥.(10%) (b)Find the vector z in Null(A) making ||z-[1 1 1 1 ]^T|| the smallest.(10%) (Hint: It is easier to use the result of (a).) (c)Find the set of all real 4x4 matrix A=A^T having the above Null(A).(10%) (Hint: Recall the Spectral Decomposition Theorem.) (注:下面以"€"代替"属於"的那个数学符号因为我找不到orz) (d)With the above Null(A), let any v€R^4 be uniquely decomposed as v=u+w, where u€Null(A) and w€Null(A)^⊥, and define a linear operator T:R^4→R^4 as T(v)=u-w. Show that T is orthogonal. (10%) Also determine all eigenvalues of T.(5%) (Hint: Consider the proper equivalent condition for the orthogonality of T.) 3.Let V be the real vector spaces with vectors [p1(x) p2(x)]^T, where p1(x) and p2(x) are polynomials in x with degrees no larger than 2. For vectors [p1(x) p2(x)]^T and [q1(x) q2(x)]^T in V, [p1(x) p2(x)]^T+[q1(x) q2(x)]^T =[p1(x)+q1(x) p2(x)+q2(x)]^T. For r€R, r[p1(x) p2(x)]^T=[rp1(x) rp2(x)]^T. (a)Find a basis B of V.(10%) (b)Consider the linear operator D: V→V defined by D([p1(x) p2(x)]^T) =[dp1(x)/dx dp2(x)/dx]^T. Find [D]B(注:B是下标) and decide if D is one-to-one or onto.(15%) -- ▇▃▁_＿◢◢ ▋▎◤ ◤◢▊ ◤▆▄ _ / ▊▉▎ │ ▅▄▄▇ ▋▏▊/▎▊◣▏ ▋● ▃◣ ▎｜▉▍ │ ▊▎▊/◢ ▃▄ ▍▃◤ ∕▊ ▊▎▎_ An apple a day ◥▊ ▎ ▂▁ ▄▅◤ /▎/ ▋/▉▊◣ ＼＼ /▍ ◥ ▋ ◤▎ ◢◤◢∕ ▉▍ ▊‵◥ ﹨﹨ keeps the doctor away◤/▎◢=◣◥█▎ ▃◢◤ ∕ ▉▏▍ ▎ ◥◥ ψmaxint --

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