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.