Linear algebra: from the inner product to the dot product

Question

Let $W$ be a $n$-dimensional vector space. What do we mean when we say that a linear transformation $f:W\to \mathbb{R}^n$ carries the inner product on $W$ to the dot product on $\mathbb{R}^n$? Is there such a linear transformation? Can you give me an example?

Answer 1

The meaning of this is the following. Imagine you have two vectors $v,w \in W$ and you want to compute their inner product. Then a map $f$ that carries the inner product on $W$ to the dot product on $\mathbb{R}^{n}$ allows you to compute the inner product of $v$ and $w$ as the dot product of the vectors $f(v) , f(w) \in \mathbb{R}^{n}$. The tipical example is the map $f:W \to \mathbb{R}^{n}$ which assigns to a vector $v$ its column of coordinates $f(v)$ $$ f(v) := \left( \begin{array}{c} v_1 \\ v_2 \\ \vdots \\ v_n \end{array} \right) $$ w.r.t. an orthogonal basis $B=(b_1,b_2,\cdots,b_n)$ of $W$. It is easy to show that orthogonal basis do exists. Indeed, take a unit vector $b_1$, then take a unit vector $b_2$ in the orthogonal complement of $b_1$, then take a third one in the orthogonal complement of the first two, and so on.