## 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:

\[\langle \alpha \mathbf{v}_1, \mathbf{v}_2 \rangle =\langle \mathbf{v}_1, \alpha \mathbf{v}_2 \rangle = \alpha \langle \mathbf{v}_1, \mathbf{v}_2 \rangle\]