MATH 219 Final, Fall 2002

ERICAN ©XNIVERSIT? LIBRARY OF BEMUT American University of Beirut N. Nahlus Mathematics 219 Final Examination (Time: 2 hrs.) Jan 28, 2003 1. (40%) Prove (concisely) or disprove (by a counter example) any 10 of the following 12 parts (in an inner product space Vv) (@) 4@,»)=le tv -le -vil (b) Non-zero orthogonal vectors are linearly independent. (c) Two mxn matrices have the same null space iff they have the same row space. (d) Similar nxn matrices have the same row space. (e) Similar matrices have the same eigenvalues. (f) Ifa set of 4 vectors (say in RS) have the property that each 3 of them are linearly independent, then this set is linearly independent. (g) f(A@B)=f(A® f(B) (for every linear transformation f & A and B are subspaces of V) (h) For an mxn matrix A, A & RRE(A) have the same null space & row space. (i) Foran mxn matrix A, A & RRE(A) have the same column space. (Gj) A 4x4 matrix with 4 distinct eigenvalues is diagonlizable. (k) There exists a square matrix with 4=0 as an eigenvalue . 1) For any §) etric nxn matrix A, A and Al have the same null space. (Hint: Prove it) y symm 2. (8%) Find the least squares (best possible) solutions of the system {x + y+2Z= 0 & xtytz=8}. 3. (8%) (a) What do we know about arbitrary symmetric matrices regarding eigenvalues & diagonalization ? (b) Apply the Cauchy-Schwarz inequality on C[a, b]= The inner product space of continous functions on the interval [a, b] (with the well-known way of “dotting” functions). 4. (7 %) Let T: V_>W be a linear transformation of vector spaces. State the rank-nullity theorem for T. Then use it to show that if T is injective and dimV=dimW=n, then T must be onto. 5. (7%) Let T:V> W bea given injective linear transformation of vector spaces (say dim V=4, dim W=6) Find a linear transformation S:W >V such that SoT=I (identity on V). (Hint: Apply the Linear Extension Theorem with good choices of bases) 3°10 6. (13%)Let A=]}6 2 0 00 5 (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. @onot calculate. P™'). 101 5 1 0 7. (12%) Let A=]2 1 4 11] & RRE(A)=|0 1 0 0 0 1 2 (i) Find a basis for each of the row space, column space, and the null space of A. Justify (ii) Does the system AX = Bhave a solution for every Bin R°? Explain. 8. (5%) Prove or disprve (by a counter example): similar nxn matrices have the same row space & the same column space.