Inner Product

UNDER CONSTRUCTION

Consider some vectors in a vector space \( V \) over \( \mathbb{F} \). We would like to define some concepts such as length of a vector or angle between two vectors, both of which are not intrinsic properties of the space itself. We notice that scalar multiplying a vector \( \vec{v} \) by a factor of \( c \) multiplies its length by \( c \), and forming the sum \( \vec{v} + \vec{w} \) causes the norm to change from \( \sqrt{\sum_i v_i^2} \) to \( \sqrt{\sum_i v_i^2 + \sum_i w_i^2 + 2 \sum_i v_i w_i} \). Let us abstract the concept. Such a function \( L: V \times W \to \mathbb{F} \) taking in two inputs has the property of being linear with respect to each input; that is, \( L(c \vec{v} + \vec{a}, d \vec{w} + \vec{b} ) = cdL(\vec{v}, \vec{w} ) + cL(\vec{v}, \vec{b}) + dL(\vec{a}, \vec{w}) + L(\vec{b}, \vec{w}) \). For a function of \( n \) inputs linear in each input, this function is called \( n \) linear. For \( n =2, V = W \), such a function is called an inner product. Inner products will be used to develop the ideas of the magnitude of a vector and the angles between two vectors in Euclidean spaces as well as some other more abstract ideas.

Let \( g: V \to V \to \mathbb{F} \) be an inner product. Choose the basis \( \mathfrak{B} = \{ \vec{e}_{i=1, ..., n} \} \) in \( V \) and define the matrix \( G = G(i, j) = g(\vec{e}_i, \vec{e}_j), 1 \leqslant i, j \leqslant n \). \( G \) is called the Gram matrix of the inner product. Then to compute the inner product of two vectors \( \vec{v}, \vec{w} \), first compute the coordinates \( [\vec{v}]_{\mathfrak{B} } \) and \( [\vec{w}]_{\mathfrak{B}} \), then calculate \( g( [\vec{v}]_{\mathfrak{B} } , [\vec{w}]_{\mathfrak{B}} ) = [\vec{v}]_{\mathfrak{B} }^t G [\vec{w}]_{\mathfrak{B}} \). This notation also makes the idea of multilinearity given in the motivation more natural. Moreover, if a basis \( \mathfrak{B} \) with \( n \) vectors is chosen and \( G \) is some \( n \times n \)matrix over \( \mathbb{F} \), then the map \( g([\vec{v}]_{\mathfrak{B}}, [\vec{w}]_{\mathfrak{B}} ) = [\vec{v}]_{\mathfrak{B}}^t G[\vec{w}]_{\mathfrak{B}} \) defines an inner product on the space spanned by \( \mathfrak{B} \) (the 2-linearity is obvious by the definition of matrix multiplication). It follows that there is a bijection between inner products on a this space of \( \dim n \) and \( n \times n \) matrices.

Let us now discuss how \( G \) is affected by a change of basis. Let \( S \) be the change of basis matrix so that \( [\vec{v}]_{\mathfrak{B}} = S [\vec{v}]_{\mathfrak{A}} \). Then the following is obtained: \( g([\vec{v}]_{\mathfrak{B}}, [\vec{w}]_{\mathfrak{B}}) = [S\vec{v}]_{\mathfrak{A}}^t G [S\vec{w}]_{\mathfrak{A}} = [\vec{v}]_{\mathfrak{A}}^t (S^t G S) [\vec{w}]_{\mathfrak{A}} \) so that the Gram matrix in the basis \( \mathfrak{A} \) is \( S^t G S \).

Exercise. Prove that \( g^t (\vec{v}, \vec{w} ) = g (\vec{w}, \vec{v} ) \).

Kostrikin, A., and Manin, Y. Linear Algebra and Geometry.