MATH 219 Final, Spring 2001

LIBRARY at mm a tat AMERIC ON PATE TRS ER, OF BEIRUT Prof. N. Haidar FACULTY OF ARTS & SCIENCES . A. U.B,... Sec. 1 Math 219 June 20, 2001 (2-nd Semester , 2000-2001) Time : 2 hours FINAL EXAM (closed book) @1. Suppose that V is a finite-dimensional vector space and S & Vv. Apply the basis extension theorem to prove that 4 a subspace T of V such thatS AT = {0} andS+T = V @ll. Let T: V > W bea linear transformation. Show that if 4,,4,,...,4, €V such that T(A,). TC A,), T(A,) are LID, then A,.A,,.....,A, are LID. en @III. Let T be a linear wransformation on a finite-dimensional vector space V. If kerT © imageT = {0}. show that V = kerT + imageT. @IV. Given V=(V. <>). with V a finite-dimensional vector space. Prove that an orthogonal set of nonzero vectors in V is LID. @V. Let T: R° > R™ be such that TOxy) = (y-x . ytx), V(X y) € R. (i) Show that T is an isomorphism. “2 -~n. . T° is involutory , ie. (it) Find a formula for Tht prove that 2 eT AL Wn) ZH lO 1.23 ce fh @VI. (i) Determine the cigenspace associated with the smallest eigenvalue of A when a -b 0 . 0 90 0 ao-b 0 (ii) prove that 0 Q —b 0 @VIIL. Prove that if V=(V. <>). with V a finite-dimensional vector space and if W¢ V, then WOW =V. OVIII. Consider the vector 5 = (60.12) € R®. Let W be the subspace of R° spanned by A, =(1.0.1) and A,= (2,1,0). . : . = a 4 G) Decompose § into the sum of a vector BE Wanda vector Ce Wo. (ii) Find the projection matrix for W and use it to find the projection of S on W. Gii) Apply the Gram-Schmidt process to construct an orthonormal basis for W. @IX. Given the linear algebraic system of equations (2 | 3 -\ a = j4 2 y (B° -8) B (4) Determine the value of B for which the system becomes inconsistent. Gi) Find the minimal norm solution (the pseudo solution) of the previous Inconsistent system.