Question

4. (a) Consider the vector space consisting of vectors where the components are complex numbers. If u = (u1, u2, u3) and v = (V1,V2, us) are two vectors in C3, show that where vi denotes the complex conjugate of vi, defines a Hermitian (compler) inner product on C3, i.e. 1· 2· 3, 4, (u, v) = (v, u), (u+ v, w)=(u, w)+(v, w), (cu, v) = c(u, v), where c E C is a constant, (u,u) > 0 and (u, u) = 0 if and only if u = 0. (b) Find the norm of the vector u = (1, 2 + i,-i).

How does one solve this problem?

Answer 1

So prove < u, v > = u_1v_1^bar + u_2v_2^bar + u_3v_3^bar satisfies all the four properties < u, v > = < u, v^bar > < v, u^bar > = (v_1u_1^bar + v_2u_2^bar + v_3u_3) = u_1v_1^bar + u_2v_2^bar + u_3v_3^bar) = u_1v_1^bar + u_2v_2^bar + u_3v_3^bar = < u, v > < u + v, w > = < u, w > + < v, w > l. h s. < u + v, w > = (u_1 + v_1) (w_1^bar) + (u_2 + v_2) (w_2^bar) + (u_3 + v_3) (w_3^bar) = u_1w_1^bar + v_1w_1^bar + u_2w_2^bar + v_2w_2^bar +u_3w_3^bar + v_3w_3^bar = (u_1w_1^bar + u_2w_2bar + u_3v_3^bar) + (v_1w_1^bar + v_2w_2^bar + v_3w_3^bar) = < u, w > + < v, w > = RHS < u, v > = c < u, v > where c is constant. < w, v > = c_1u_1v_1^bar + u_2v_2^bar + c_3v_3^bar = c[u_1v_1^bar + u_2v_2^bar + u_3v_3^bar] = c < u, v > = R.H.S