Question

Upts) GIve the text of the Spectral Theorem on a real inner product space E (3pts) Prove that any eigenvalue of a self-adjoint linear map on a complex inner product space is real. 4,) (3pts) Give the definition of a skew-symmetric matrix. X Lexercisebethe car points baseofPandaERaparameter -C )ER . For all = ( 1 ) E R3 and y-(h /2 yE R2 we define the bilinear form ba by 4 y. (3pts) For which value of a, b, is it symetric? Xp. (3pts) For which values of , ba is it definite positive ( we can use Gauss's method)? 3. (2pts) Let pa(x, y) (ba(,y) + baly,x)). Prove that pa(x, y) 81y1- (2+ a)(z1y2 + 2h) +6z2/2 4. (3pts) What is the quadratic form ga associated to pa 5. We note q 9-10, (? -10), (a) (3pts) What is the signature of q? (b) (4pts) Give a q- orthogonal base for R2 Exercise 3 (18 points)

Answer 1

Please mention the question number which you are asking for. Only 4 question was circled so i answered the 4 ques. If you need solution of another one please repost the question and mention the question number. Thank you.

Solution → on ative , i e it stis -3 +8D -03 -3 8o