(b) (d) Let ( , ) be an inner product on R". Prove that there is a matrix A M,,,(R) such that (2, 7)=Z2TAY forall T, 7 R The matrix A is called the standard matrix of { , ). Your proof should indicate what the (i,j)th entry of A is. [Hint: ( , ) is a bilinear form on R".] Find the standard matrix of: (i) The dot product on R", (ii) The inner product on R? defined by (I, 7} = z1y1 + 22192 + 27251 + Hz2ye. Let A be the standard matrix of an arbitrary inner product { , ) on R". Prove: (i) A is a symmetric matrix. [Reminder: Symmetric means AT = A] (ii) The eigenvalues of A are strictly positive real numbers. Let a,b,c,d R. Prove that (T, ) = az1yr + bx1y2 + cxayy + daays defines an inner product on R? if and only if b=, a>0 and ad?b >0