how inner products are defined on a vector space? 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.

Every positive-definite matrix $M$ is associated to an inner product when viewed as a quadratic form. In this case the matrix is $M=\begin{pmatrix}1&-1\\-1&4\end{pmatrix}$ and the inner product is given by $\langle x,y\rangle=x^TMy$.
