## The Inner Product on a Vector Space

An inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. The dot product is a vector space, as is the magnitude of the cross product of two vectors. If
$\mathbf{v}_1, \: \mathbf{v}_2$
are vectors in a vector space
$V$
and
$T$
is a linear transformation sending elements of the space onto other elements, with associated matrix
$M$
, then
$\mathbf{v}^T_1 M \mathbf{v}_2$
is the inner product of
$v_1$
with
$v_2$
.
$T$
may be any matrix, including the zero matrix.
The inner product is symmetric:
$\langle \mathbf{v}_1, \mathbf{v}_2 \rangle =\langle \mathbf{v}_2, \mathbf{v}_1 \rangle$

The inner product is positive definite:
$\langle \mathbf{v}, \mathbf{v} \rangle \geq 0$
and
$\langle \mathbf{v}, \mathbf{v} \rangle = 0 \leftrightarrow \mathbf{v}= \mathbf{0}$

The inner product is linear in both arguments:
$\langle \alpha \mathbf{v}_1, \mathbf{v}_2 \rangle =\langle \mathbf{v}_1, \alpha \mathbf{v}_2 \rangle = \alpha \langle \mathbf{v}_1, \mathbf{v}_2 \rangle$