Inner Product Spaces

We had learned that given vectors ${\vec i}$ and ${\vec j}$ (which are at an angle of $90^{\circ}$ ) in a plane, any vector in the plane is a linear combination of the vectors ${\vec i}$ and ${\vec j}.$ In this section, we investigate a method by which any basis of a finite dimensional vector can be transferred to another basis in such a way that the vectors in the new basis are at an angle of $90^{\circ}$ to each other. To do this, we start by defining a notion of INNER PRODUCT (dot product) in a vector space. This helps us in finding out whether two vectors are at $90^{\circ}$ or not.