1、EXAMINATION OF LINEAR ALGEBRAClass: Name:Number : Score:1 Choose the best answer (24 points)1. If A is m n and b is nonzero m column vector with m:x1 ax2 2x3 = 1x1 x2 +ax3 = 25x1 5x2 4x3 = 1Determine the conditions for a such that the system is consistent and inconsistent. More-over, nd all solution
2、s for consistent system. (8 points)419. Find an orthonormal basis for R4 consisting of eigenvectors of the matrix0BBBBB1 1 1 11 1 1 11 1 1 11 1 1 11CCCCCA:(8 points)520. Let W = SP(S) be a subspace, whereS =8:0BB1121CCA;0BB1011CCA;0BB2131CCA;0BB0111CCA;9=;Find a subset T of S such that T is a basis
3、for W and determine dim(W): (8 points)21. Let A be an n n positive de nite matrix.(1). Prove that A 1 is positive de nite matrix.(2). Prove that B =A AA A!is positive semide nite matrix, but is not positivede nite. (10 points)622. Let A = (aij) be an n n matrix. Denote by A = (A1; ;An) and I A =(B1; ;Bn). If A2 = A, then(1). All eigenvalues of A are either 0 or 1.(2). SPfA1; ;An;B1; ;Bng= Rn.(3).prove that A is diagonalizable. ( 10 points).7
