How do mathematicians define inner product on a vector space.
For example: $a = (x_1,x_2)$ & $ b =(y_1,y_2) $ in $ \mathbb{R}^2.$
Define $\langle a,b\rangle= x_1y_1-x_2y_1-x_1y_2+4x_2y_2$. It's an inner product.
But how does one motivate this inner product? I think there is some sort of matrix multiplication between some vectors.
Best Answer
The abstract definition of an inner product of real-valued vectors is a function $\langle \, , \rangle: \mathbb R^n \times \mathbb R^n \to \mathbb R$ satisfying the following axioms, where $\alpha$ and $\beta$ are scalars and the $x$'s and $y$'s are vectors.
Conditions (1) and (2) tell us that an inner product should be linear in both its arguments. In addition, condition (2) is not strictly necessary, as conditions (1) and (3) imply it. This definition also extends to arbitrary vector spaces, not just over $\mathbb R$.