MATH 218 Final, Fall 2002

ERICS\ UNTVERSIT" LIBRARY OF BEIRUT Mathematics 218 Final Exam. N. Nahlus Time: 2hrs. Jan 25, 2003 1. (24%) Prove (concisely) or Disprove (by a counter example) in an inner product space. (a) ut vl tlle v= 2 lel? 42111 (b) If llulliivlle<u.v> &u,v are non — zero then u & vare not orthogonal (c) Non-zero orthogonal vectors are linearly independent. (d) If two mxn matrices have the same row space, then they have the same_rank and nullity. (e) If two mxn matrices have the same row space, then they have the same null space. (f) If Ais an orthogonal nxn matrix then detA=+1 (g) (For any square matrix A), A and A® have the same row space. 2. (20%) (a) Find the least squares solutions of the system {x + y =0 & xt+y=l &xt+y= 4} (b) What do we mean precisely by a least square (best possible) solution ofa non-consistent system AX =5? (c) What do we know about arbitrary symmetric nxn matrices regarding eigenvalues & diagonalization ? (d) Apply the Cauchy-Schwarz inequality on the continous functions on the interval [0, 3]. (e) Write the orthogonal projection formula for a vector a on a subspace W (given an o.n basis of W). 3 6 0 3. (15%) Let A=!|1 2 0 005 (i) Find the eigen values of A. and a basis for each eigen space of A. (ii) Show that A is diagonalizable and_find the exact relation between A,P and D. (Do not calculate P™). 4. (9%) Let T: V_—>W be a linear transformation of vector spaces. If T(a), T(ao), »-+> Tan) are linearly independent and dimV=n, show that {a1, a, ---) 4a jis a basis of Vv. 5, (9%) Let T: V>W be a linear transformation of vector spaces. (i) State the rank-nullity theorem for T, (ii) then use it to show that if T is onto and dimV=dimW=n, then T must be is injective. 1 1638 6. (9%)Let A=|2 4 0 2 2 2 001527 (i) Show that rank A=2 (ii) Does the system AX =B have a solution for every B in R°? Justify. 7. (9%) Let { by, Das os Da 5 Bt, B25 e+-9 Am } bean on basis of an inner product space V. Let B= span { bj, bz, -.-, b,} & A=span {@1, 82, ++) Am }. Show that (i) V=A@B (ii) B=A*. 8. (5%) For any symmetric nxn matrix A, show that 4 and A have the same null space. a 7