The usual inner product (also called "dot product") between two vectors is multiplying the components and then summing the results. For example, the inner product of $ \begin{bmatrix}2 & 3 & 4\end{bmatrix} $ and $ \begin{bmatrix}1 & 0 & 5\end{bmatrix} $ is $ 2*1+3*0+4*5=22 $.
Note that the inner product may be defined differently, but this is the usual defintion. In some contexts, the vector space is assumed to be all of $ R^p $.
As an inner product, it satisfies several properties.
- $ x\cdot{x}\ge{0} $ for all vectors in the vector space.
- $ x\cdot{x}=0 $ only when x=0.
- $ x\cdot{y}=y\cdot{x} $ (ie it is commutative)
- $ x\cdot{y+z}=x\cdot{y}+x\cdot{z} $ (and thus $ (x+y)\cdot{z}=x\cdot(y)+x\cdot(z) $)
