MATH 218 Final, Summer 2002

Le red ” °-Nfathematics 218 Final Exam. (2 hrs.) August 26, 2002 1. (24%) Prove (concisely) or disprove (by a counter example) in an inner product space. fa) dda slaty? wave (b) If feeb snc) then wee. (c) Non-zero orthogonal vectors are linearly independent. (dQ) Tf two asx matrices have the same aull space. then they have the same row space. (e) TP two ae matrices have the same null space, then they have the same column space. (fF) (For any square mates A). 4 and -f! have the same null space. 2. (10%) (a) What do we mean precisely by a least square solution of a non-consistent system I ¥ by | ys AVY =h?) AND in) how do we find them? (b) What do we know about symmetric 7x matrices regarding eigenvalues & diagonalization ? (c) Apply the Cauchy-Schwarz inequality on functions in Cla. b] & vectors in R”. 3. (15%) Let (i) Find the eigen values of A and a basis for cach eigen space of A, (i) Show that A is diagonalizable and_find the exaet relation between A and its diagonalization D. (Yes. calculate P7!). (6%) Let TP: VW be a linear transtormation of vector spaces. If T(a;}, Taz), ..., T(a,) are inearly independent. stow that fay. as. .... a, }are linearly independent. 5. (6%) Let T: VW bea linear transformation of vector spaces. If Tas 1-1 and dimV=dimW=n, show that Tis onto. (Hint: You may use the rank-nullity theorem. 6. (Mo) Let Wy We Wa dis cis. os ae | bE an o.n basis of an inner product space V. het Wo span fw). we... wad and let A=span fay.ay, 2... a) Show “sat. Then deduce that F = IF @W~. 7. (6%) IP fac boc} is a basis of a vector space V. show that fat+b. u+2b. btc} is also a basis of V. lau | (1) Show that rank A=2 (i) Prove or disprove that the first 2 rows of A forma basis for the row space of A (i) Does the system LV = 8 have a solution for every Bin R*? Justify. 9. (6%) Suppose dimV=dimAtdimB and ao 8 =0 where A and B are subspaces of V. Show that V=A+B. 10. (5%) For any rectangular matrix A. show that land 4/f have the same column space.