What is an orthonormal basis relative to the dot product What is an orthonormal basis relatively to the dot product. I know that an orthonormal basis is a basis who's vectors are orthogonal to each other and that are unit vectors but I'm not sure what it means that the basis is relative to the dot product.
 A: It's not the basis that is relative to the dot product but its orthonormality. The concept of a basis exists in general vector spaces. However, in general vector spaces there is no concept of orthogonality or of a norm. Both concepts can be defined via a dot product, namely $\vec a$ and $\vec b$ are orthogonal if $\vec a\cdot\vec b=0$ and the norm of $\vec a$ is $|\vec a|=\sqrt{\vec a\cdot\vec a}$. These concepts of orthogonality and norm are said to be induced by the dot product. An orthonormal basis relative to a dot product is a basis whose vectors are normalized with respect to the norm induced by the dot product and are pairwise orthogonal with respect to the concept of orthogonality induced by the dot product.
For instance, in $\mathbb R^n$, the usual concepts of orthogonality and norm are the ones induced by the usual dot product, $\vec a\cdot\vec b=\sum_ia_ib_i$. If you introduce a different dot product, e.g. $\vec a\cdot\vec b=a_1b_1+2a_2b_2$ in $\mathbb R^2$, this induces different concepts of orthogonality and norm, and different bases will be orthonormal than under the canonical definitions.
