Lecture 27 - Starting Inner Product Spaces

Today we'll briefly start chapter 6, which covers inner product spaces.

Introduction to Direction and Magnitude

When we started talking about vectors, usually we talk about vectors having direction and magnitude. Previously, we just added and scaled these vectors, but we haven't talked about these other quantities. We now will try to abstract what these quantities mean, and how to derive them for general vector spaces.

In $R^{n}$ , we already have a notion of length. If $\vec{x} = (x_{1}, . . ., x_{n})$ then the magnitude of $\vec{x}$ is:

| | \vec{x} | | = \sqrt{x_{1}^{2} + \dots + x_{n}^{2}}

you may see these as single bars or double bars. Both are the same. In $n$ -dimensions this is the classic usage of the Pythagorean Theorem. But recall that:

x_{1}^{2} + \dots + x_{n}^{2} = \vec{x} \cdot \vec{x}

Thus:

| | \vec{x} | | = \sqrt{\vec{x} \cdot \vec{x}}

Now notice that this is the square root of a "dot" product. What if we worked with $C^{n}$ instead? Can we extend this idea of a dot product? Here:

\vec{v} = [\begin{matrix} z_{1} \\ z_{2} \\ ⋮ \\ z_{n} \end{matrix}]

Notice if we just check each $x_{i}$ as $z_{i}$ then that may lead to the dot product being negative, resulting in a complex magnitude. That seems weird to have, so instead we use:

| | \vec{v} | | = \sqrt{| z_{1} |^{2} + \dots + | z_{n} |^{n}}

But now this is NOT the dot product with itself anymore. Recall that since $| z |^{2} = z \cdot \bar{z}$ so then:

| | \vec{v} | | = \sqrt{z_{1} {\bar{z}}_{1} + \dots + z_{n} {\bar{z}}_{n}}

As such, we make the following definition:

Definition

Define:

[\begin{matrix} z_{1} \\ z_{2} \\ ⋮ \\ z_{n} \end{matrix}] \cdot [\begin{matrix} w_{1} \\ w_{2} \\ ⋮ \\ w_{n} \end{matrix}] = z_{1} {\bar{w}}_{1} + \dots + a_{n} {\bar{w}}_{n}

where $z_{i}, w_{i} \in C$ .

Here notice that now the commutativity of the dot product is lost. Now we want to think about the notion of length in an abstract space $V$ .

Length in an Abstract Space $V$

We pick out the properties that are important to define lengths and distances in this way.

inner product

An inner product on a vector space $V$ is a function (binary operator) $\cdot : V \times V \to F$ satisfying the following (where $⟨ \vec{u}, \vec{v} ⟩$ is the inner product of $\vec{u}$ and $\vec{v}$ ):

(Positivity): $⟨ \vec{v}, \vec{v} ⟩ \geq 0$ for all $\vec{v} \in V$ ;
(Definiteness): $⟨ \vec{v}, \vec{v} ⟩ = 0$ iff $\vec{v} = \vec{0}$ ;
(Additivity in 1st Slot): $⟨ \vec{u} + \vec{v}, \vec{w} ⟨ = ⟨ \vec{u}, \vec{w} ⟩ + ⟨ \vec{v}, \vec{w} ⟩$ for all $\vec{u}, \vec{v}, \vec{w} \in V$ ;
(Homogeneity in 1st Slot): $⟨ λ \vec{u}, \vec{v} ⟩ = λ ⟨ \vec{u}, \vec{v} ⟩$ where here $λ \in F$ for all $\vec{v}, \vec{u} \in V$ ;
(Conjugate Symmetry): $⟨ \vec{u}, \vec{v} ⟩ = ⟨ \vec{v}, \vec{u} ⟩$ for all $\vec{v}, \vec{u} \in V$ .

Notice why we give the definition the way it is:

We need positivity so that when we take its square root, we always get a number over a field like $R$ or $C$ . Actually, notice that we need orderedness here, so we really are only restricting ourselves to $R$ .

Examples of Inner Products

The important thing here is that there are inner products that are not the dot product. The dot product is a specific case of this more general one.

The Dot Product

The dot product defined via Lecture 27 - Starting Inner Product Spaces#^2bbc0d can be shown to follow all our properties of the inner product. As an example, for property 2, we require that the number under the square root be 0, so since all summands are positive then all $z_{i} {\bar{z}}_{i} = 0$ which is only true if $a^{2} + b^{2} = 0$ so then $a, b = 0$ so $z_{i} = 0$ as required.

The Weighted Dot Product

Another example of an inner product is the weighted dot product on $C^{n}$ defined by:

⟨ (x_{1}, . . ., x_{n}), (y_{1}, . . ., y_{n}) ⟩ = c_{1} x_{1} {\bar{y}}_{1} + \dots + c_{n} x_{n} {\bar{y}}_{n}

where all $c_{i} \in R^{+}, x_{i}, y_{i} \in C$ .

The Integral Product

One last example, suppose $C (R)$ is the space of continuous functions on the interval $(- 1, 1)$ . Notice that this is a real vector space specificallly. We define:

⟨ f, g ⟩ = \int_{- 1}^{1} f (x) g (x) d x

is an inner product. As an example, let's look at property 1. If $f = g$ then we are integrating over $f^{2}$ which is always positive, proving (1). As an instance using this:

⟨ - x, x^{2} + 1 ⟩ = \int_{- 1}^{1} (- x) (x^{2} + 1) d x = \int_{- 1}^{1} (- x^{3} - x) d x = 0

since we have an odd function.

A special property if our inner product gives a value of $0$ is that these vectors are orthogonal, or perpendicular, to each other. Similarly, we can get the length of a vector in a similar way:

⟨ x, x ⟩ = \int_{- 1}^{1} x \cdot x d x = \frac{2}{3}

Introduction to Direction and Magnitude

Length in an Abstract Space V

Examples of Inner Products

The Dot Product

The Weighted Dot Product

The Integral Product

Length in an Abstract Space $V$