Scalar Product on $\mathbb{R}^3$

Question

I have a question on how to solve the following problem.

Find a scalar product on $V=\mathbb{R}^3$ such that $\{e_1, e_2, e_3\}$ defined as $e_1=(1,1,0)$, $e_2=(1,0,1)$, $e_3=(0,1,1)$ is an orthonormal basis.

I think I have to find a bilinear form $\varphi$, such that $\varphi(e_i,e_j)= \delta_{ij}$ (for the vectors to be orthogonal) and $\|e_i\|=1$.

How can I find this $\varphi$?

Answer 1

Take a positive definite symmetric matrix

$$A:=\begin{pmatrix}a&x&y\\x&b&z\\y&z&c\end{pmatrix}.$$

You want

$$(1,1,0)A\begin{pmatrix}1\\1\\0\end{pmatrix}=1\iff a+b+2x=1.$$

Similarly, you obtain two more equations for the other two vectors. Furthermore, you can obtain six equations by the orthogonality relations. Then choose a suitable $(a, b, c, x, y, z)$, which fulfills all equations.

Answer 2

Treating the basis vectors $e_k$ as column vectors, you’re essentially looking for a symmetric positive-definite matrix $G$ such that $e_i^TGe_j = \delta_{ij}$. If we assemble these into a matrix $P$, we can collect all of these equations into the single “bulk” equation $$P^TGP=I.$$ The $e_k$ are linearly independent, so $P$ is invertible, therefore $G=P^{-T}P^{-1}$.