Does Gram-Schmidt always work correctly?

Question

This question is from E. Linear Algebra by Larson 7ed.

Use the inner product $\langle u,v\rangle = 2u_1v_1 + u_2v_2$ in $\mathbb{R}^2$ and the Gram-Schmidt orthonormalization process to transform ${(2,-1),(-2,10)}$ into an orthonormal basis.

I solved the question and I got a non-orthonormal basis!

Actually, the next question asks why the result is not orthonormal when the Euclidean inner product on $\mathbb{R}^2$ is used?

I verified that this inner product is valid for all the properties of the inner product but I can not think of an answer to this question! Please help.

Answer 1

With the usual inner product in $\,\Bbb R^2\,$ :

$$v_1=(2,-1)\;,\;\;v_2=(-2,10)\implies u_1:=\frac{v_1}{||v_1||}=\frac1{\sqrt5}(2,-1)$$

$$e_2:=v_2-\left(u_1\cdot v_2\right)u_1=(-2,10)+\frac{14}5(2,-1)=\left(\frac{18}5\,,\,\frac{36}5\right)$$

Then just define

$$u_2=\frac{e_2}{||e_2||}\;,\;\;\text{and check}\;\;\{u_1,u_2\}\;\;\text{is an orthonormal basis}$$