Question

be an inner product on R". In this question you will show that there exists a n x n Let matric A such that (3.1) for all ,je R". Further you will show that A must be symmetric. (1) Suppose that such a matrix exists. That is suppose that there exists anxn matrix A = (a;^) for which (3.1 holds. Calculate (ē;, ë;) (2) Describe A in terms of (ēi,e;) 1 < i, j < n. (3) Prove that there exists an n x n matrix A for which (3.1) holds. (4) Show that A in (3.1) is symmetric, i. e., AT = A (5) Classify all the inner products in R2.2 in terms of entries of A

Please give me right answer with the details. Thanks!

Answer 1

- [arsJ prn A upbone Ar T < that u ch shwa And 0] Cac) e じA 1,0, 1 Cth i th 20u A ci A ith aow AJ O Sth 1-- aui O. aij 1 2). bart ) By [<ei,elm Lai 1 nAn nn.

I4 C) Ah [ <ci, ci>J een 3). Let A = whes rocuct On inner an < R, 37= Nou. n I & ]+ 1 n <e, e, 7 <cez7 ne <ce, en7 Jn LE1 nAS Ard han n ni'<ciei7 n f sW.

57 A3 ) ) BY 4). 3). By. uch tat pro duct 7 Inner in ac C, n <ei, ej 7 = <e, cr7 1 <C), ci 2 = aji o n Fre duet innn T A n A EA AeR w hu