# Linear Algebra - Inner product of two vectors

Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product).

An inner product space is a vector space with an additional structure called an inner product.

Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product) to vector spaces of any (possibly infinite) dimension.

The Inner product of vectors u and v is written: $\left \langle u,v \right \rangle$

## Definition

For the real numbers and complex numbers, the definition of the inner product may be flexible.

This flexibility is used heavily, e.g. in Machine Learning

### Dot product

In Euclidean spaces (vectors over R?), the inner product is the dot-product.

This definition leads to the norm of a vector being the geometric length of its arrow.

$$\left \langle u,v \right \rangle = u.v$$

## Property

• linearity (in the first argument):$\left \langle u + v, w \right \rangle = \left \langle u, w \right \rangle + \left \langle v, w \right \rangle$
• symmetry: $\left \langle u, v \right \rangle = \left \langle v, u \right \rangle$
• homogeneity:$\left \langle \alpha.u, v \right \rangle = \alpha . \left \langle u, v \right \rangle$

When the inner product definition is the dot-product definition, these properties are easy to prove.