Use the inner product u v 2u1v1 u2v2 in R2 and the GramSch

Use the inner product u, v = 2u1v1 + u2v2 in R2 and the Gram-Schmidt orthonormalization process to transform {(2, 1), (8, 4)} into an orthonormal basis.

Solution

First we transform intoa orthogonal basis and then orthonormal basis

v1=(2,-1)

v2=(8,4)

|v1|=sqrt{5}

v1.v2=12

Hence, v2\'=v2-(v1.v2/|v1|^2)v1=v2-(12/5)v1=(8,4)-12(2,-1)/5=(8,4)-(24/5,-12/5)=(16/5,32/5)

v1,v2\' form an orthogonal basis

Now we normalize them

u1=v1/|v1|=(2,-1)/sqrt{5}

u2=v2\'/|v2|=(1,2)/sqrt{5}