Question

I need serious help with this problem:

Show that the function G(v,w)=v1w1+2v1w2+2v2w1+5v2w2 is an inner product on R^2, where v=(v1,v2) and w=(w1,w2). Find an orthonormal basis {u1,u2} for R^2 relative to G.

Answer 1

Let's see first if G satisfies the properties of an inner product on R^2:
1)\(G(v,v)=v_1v_1+2v_1v_2+2v_2v_1+5v_2v_2=v^2_1+4v_1v_2+5v_2^2=(v_1+2v_2)^2+v_2^2\ge0.\) Note here that G=0 only at v=(0,0).

Answer 2

2) It's clear that \(G(v,w)=v_1w_1+2v_1w_2+2v_2w_1+5v_2w_2=G(w,v)\).

Answer 3

3) For a vector u=(u_1,u_2) in R^2, we have:
\(G(u+v,w)= (u_1+v_1)w_1+2(v_1+u_1)w_2+2(u_2+v_2)w_1+5(v_2+u_2)w_2\)
\(=(u_1w_1+2u_1w_2+2u_2w_1+5u_2w_2)+(v_1w_1+2v_1w_2+2v_2w_1+5v_2w_2)\)
=\(G(u,w)+G(v,w).\)

Answer 4

In a similar way you can show that \(G(cv,w)=cG(v,w)\), for c a real scalar.