Question

let U=(u1,u2) AND V=(v1,v2) belong to R2 verify that <u,v> =u1v1-2u1v2-2u2v1+5u2v2 is an inner product space on r2

Answer 1

 As we know-

$<au,v> = <a(u_1,u_2), (v_1,v_2)> = <(au_1,au_2), (v_1,v_2)> \\[9pt] = (au_1)v_1 - (au_2)v_1 - (au_1)v_2 + 5(au_2)v_2 \\[9pt] = u_1(av_1) - u_2(av_1) - u_1(av_2) + 5u_2(av_2)$

Note that $av = a(v_1,v_2) = (av_1,av_2) = ((av)_1,(av)_2)$

 

Then we have 

 

$<u,v> = u_1(v)_1 - u_2(v)_1 - u_1(v)_2 + 5u_2(v)_2$