课程名称︰线性代数一 课程性质︰必修／选修 课程教师︰林绍雄 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2012/1/7 考试时限（分钟）：180 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : There are problems A to E wit a total of 140 points. Please write down your computational or proof steps clearly on the answer sheets. ------------------------------------------------------------------------------ -1 A. (28 points) Find an invertible matrix P ∈ M(4, R) so that P AP is the Jordan form of the matrix A = ┌ 1 -1 0 -1 ┐. Write down the │ 0 2 0 1 │ │ -2 1 -1 1 │ └ 2 -1 2 0 ┘ Jordan form of A explicitly, and find its minimal polynomial. ------------------------------------------------------------------------------ B. Let A be the matrix A = ┌ 1 0 0 ┐. C(A) is the column space of │ -1 1 0 │ │ 0 -1 1 │ └ 0 0 -1 ┘ 4 A in R . (a) has 10 points, and each of (b), (c), (d), (e) has 7 points. (a) Apply the Gram-Schmidt orthogonalization process to find an orthonormal basis of C(A) from the column vectors of A, and then write down the QR-decomposition of A. T (b) Perform the LDU-decomposition for A A to obtain a QR-decomposition of A. Do you get the same answer as in (a)? (c) Write down the projection matrix of the orthogonal projection from 4 R onto C(A), and find the minimal distance of the vector T v = [1, -1, 1, 1] to C(A). (d) Find the least square approximate solution to the linear system x = 3, x - x = a, x - x = 1, x = 4, where a is a constant. 1 1 2 2 3 3 What conditions should a satisfy so that this approximate solution becomes an exact solution? + (e) Find A . (Moore-Penrose pseudoinverse), Use this matrix to redo (c). ------------------------------------------------------------------------------ C. (14 points) Find a Schur decomposition of the matrix A =┌ 3 0 2 ┐. │ 2 3 0 │ └ 0 2 3 ┘ Is this matrix normal? ------------------------------------------------------------------------------ D. Prove the following statements. Each has 8 points. (a) Let A ∈ M(n,F) be a nilpotent matrix with nullity δ. If its minimal k polynomial is x , prove that kδ >= n. (b) Let A, B ∈ M(n,F). Prove that the set of all eigenvalues of AB equals the set of all eigenvalues of BA, but AB and BA may not be similar. (c) Let A ∈ M(n,C). Prove that A is unitary iff A is normal, and every eigenvalue of A has absolute value 1. (d) Let V be a finite-dimensional inner product space over F, Its inner product is denoted by <u, v> for u,v ∈ V. If W ⊂ V is a subspace, ┴ prove that W is isomorphic to V/W. ------------------------------------------------------------------------------ E. Determine which of the following statements is true. Prove your answer. Each has 7 points. (a) Let A, B ∈ M(n,R). Then they are similar in M(n,C) iff they are similar in M(n,R). (b) Let A, B ∈ M(3,F). Then A and B are similar iff whenever A satisfies p(A) = 0 for some polynomial p(x), B must satisfy p(B) = 0, and vice versa. + (c) Given A ∈ M(n,C) with det(A) = 0. Then the nullities of A and A + + are the same, and AA = A A. 2 (d) For a given matrix A ∈ M(n,F), the number of Jordan blocks of A is the same as the number of Jordan blocks of A. -- --

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