Let u x1x2 and nu y1y2 be the elements of R2 PROVE THAT TH

Let u = (x_1,x_2) and nu = (y_1,y_2) be the elements of R^2 .PROVE THAT THE FOLLOWING FUNCTION DEFINES AN INNER PRODUCT SPACE ON R^2 (U,V) = 4x_1y_1 + 9x_2y_2

Solution

For this function to define inner product space it must satisfy following three properties

1. Symmetry

u=(x1,x2),v=(y1,y2)

<U,V>=4x1 y1+9x2 y2

<V,U>=4y1 x1+9y2 x2=4x1 y1+9x2 y2

Hence, <U,V>=<V,U>

2. Linearity in first argument

<aU,V>=<(ax1,ax2),(y1,y2)>=4 ax1 y1+ 9 ax2 y2=a(4x1 y1+9x2 y2)=a<U,V>

Hence,

<aU,V>=a<U,V>

3. Positive definiteness

<U,U>=4x1^2+9x2^2>=0

Hence proved.