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Section 36.1 Motivation

The ten Vector space axioms are designed to capture the algebraic patterns of vectors. But when we first learned about vectors by studying the “model” vector space \(\R^n\) in Chapters 12–15, we relied heavily on geometric notions and intuition, and indeed much of our focus was on geometric concepts such as norm (i.e. length), angle, and orthogonality (i.e. perpendicularity). None of these concepts is reflected in the ten Vector space axioms.

Can the geometric patterns be found in other vector spaces? Does a matrix have “length”? Can two functions be “perpendicular” to each other? What are the mechanisms by which we would measure such things? The purpose of this chapter and the chapters that follow it is to answer these questions.

In the vector space \(\R^n\text{,}\) both geometric concepts of norm and angle are connected to dot product:

\begin{align*} \unorm{v} \amp = \sqrt{\udotprod{v}{v}} \text{,} \amp \theta \amp = \inv{\cos} \left( \frac{\udotprod{u}{v}}{\unorm{u}\unorm{v}} \right) \text{.} \end{align*}

Since both quantities can be defined in terms of the dot product, it seems that looking for some sort of “dot product” in other vector spaces is the first step to transplanting these geometric concepts in those spaces. So we need to both understand what the dot product is mathematically, and what properties of the dot product are essential for its connections to geometry.

A pairing is a function that takes two inputs, both of the same kind, and outputs a number. Pairings are usually denoted with bracket notation; we will use angle brackets to avoid confusion with bracket notation for vectors in \(\R^2\text{.}\) So

\begin{equation*} \inprod{a}{b} \end{equation*}

will represent the number that results by pairing objects \(a,b\text{.}\) With this new abstract concept, we can say that the dot product is a pairing of vectors from \(\R^n\):

\begin{equation*} \uvecinprod{v}{w} = \udotprod{v}{w} = v_1 w_1 + v_2 w_2 + \dotsb + v_n w_n \text{.} \end{equation*}

The order that objects are listed in a pairing sometimes matter, so that, in general,

\begin{equation*} \inprod{b}{a} \neq \inprod{a}{b} \text{.} \end{equation*}

But we know that order does not matter in the dot product, which is important geometrically because it shouldn't matter whether we say “\(\uvec{u}\) is orthogonal to \(\uvec{v}\)” or “\(\uvec{v}\) is orthogonal to \(\uvec{u}\)”, for example. So we would like order to not matter in a pairing of vectors in an abstract real vector space.

Below we state this preference, along with other properties that we would like a vector pairing for vectors in a real vector space to have. These properties are modelled on properties of the dot product from Proposition 13.5.3, where as usual a boldface letter represents a vector and an italic character represents a scalar.

List 36.1.1. (RIP) Real inner product axioms
  1. Symmetry.

    Every \(\uvec{v},\uvec{w}\) satisfy \(\uvecinprod{w}{v} = \uvecinprod{v}{w}\text{.}\)

  2. Additivity.

    Every \(\uvec{u},\uvec{v},\uvec{w}\) satisfy \(\inprod{\uvec{u}+\uvec{v}}{\uvec{w}} = \uvecinprod{u}{w} + \uvecinprod{v}{w}\text{.}\)

  3. Homogeneity.

    Every \(k,\uvec{v},\uvec{w}\) satisfy \(\inprod{k \uvec{v}}{\uvec{w}} = k \uvecinprod{v}{w}\text{.}\)

  4. Positive definiteness.

    Every nonzero \(\uvec{v}\) satisfies \(\uvecinprod{v}{v} \gt 0\text{.}\)

In the rest of this chapter we will begin exploring the properties of vector pairings that satisfy the four properties above, and begin to explore the geometry of abstract vector spaces that have such a pairing.