Inner Product

Section 6.1 Inner Product

Goals.

We will

define an inner product,
recognize its properties and definition in old examples,
test functions to see if they are inner products,
generalize the concept of distance, and
generalize the concept of orthogonality.

Subsection 6.1.1 Definition

Definition 6.1.1. Inner Product.

A function from \(U \times U\) to the reals for a vector space \(U\) is an inner product if and only if it meets the following conditions.

\(\langle\vec{u},\vec{v}\rangle = \langle\vec{v},\vec{u}\rangle\text{.}\)
\(\displaystyle \langle\vec{u}+\vec{v},\vec{w}\rangle=\langle\vec{u},\vec{w}\rangle+\langle\vec{v},\vec{w}\rangle. \)
\(\displaystyle \langle c\vec{u},\vec{v}\rangle=c\langle\vec{u},\vec{v}\rangle=\langle\vec{u},c\vec{v}\rangle. \)
\(\langle\vec{u},\vec{u}\rangle \ge 0, \) and \(\langle\vec{u},\vec{u}\rangle=0 \) iff \(\vec{u}=\vec{0}\text{.}\)

Checkpoint 6.1.2.

Describe each property using colloquial English. Some are like algebraic properties.

Subsection 6.1.2 Identifying Inner Products

To begin understanding the definition we will consider some specific functions and check if the properties are met. These examples are based on the dot product: something we already know.

Example 6.1.3.

We consider the function \(\langle p(x),q(x) \rangle = p(0)q(0)+p(1)q(1)+p(2)q(2) \) for the space of continuous functions.

\(\langle\vec{u},\vec{v}\rangle = \langle\vec{v},\vec{u}\rangle\text{.}\) Note because of commutativity of real number multiplication

\begin{align*} \langle p(x),q(x) \rangle = & p(0)q(0)+p(1)q(1)+p(2)q(2)\\ = & q(0)p(0)+q(1)p(1)+q(2)p(2)\\ = & \langle q(x),p(x) \rangle \end{align*}

\(\langle\vec{u}+\vec{v},\vec{w}\rangle=\langle\vec{u},\vec{w}\rangle+\langle\vec{v},\vec{w}\rangle. \)

\begin{align*} \langle p(x)+q(x),r(x) \rangle = & [p(0)+q(0)]r(0)+[p(1)+q(1)]r(1)+[p(2)+q(2)]r(2)\\ = & p(0)r(0)+q(0)r(0)+p(1)r(1)+q(1)r(1)+p(2)r(2)+q(2)r(2)\\ = & p(0)r(0)+p(1)r(1)p(2)r(2)+q(0)r(0)+q(1)r(1)++q(2)r(2)\\ = & \langle p(x),r(x) \rangle+\langle q(x),r(x) \rangle \end{align*}

\(\langle c\vec{u},\vec{v}\rangle=c\langle\vec{u},\vec{v}\rangle=\langle\vec{u},c\vec{v}\rangle. \)

\begin{align*} \langle cp(x),q(x) \rangle = & cp(0)q(0)+cp(1)q(1)+cp(2)q(2)\\ = & c(p(0)q(0)+p(1)q(1)+p(2)q(2))\\ = & c\langle p(x),q(x) \rangle\\ = & p(0)cq(0)+p(1)cq(1)+p(2)cq(2)\\ = & \langle p(x),cq(x) \rangle \end{align*}

\(\langle\vec{u},\vec{u}\rangle \ge 0, \) and \(\langle\vec{u},\vec{u}\rangle=0 \) iff \(\vec{u}=\vec{0}\text{.}\)

\begin{align*} \langle p(x),p(x) \rangle = & p(0)p(0)+p(1)p(1)+p(2)p(2)\\ = & p(0)^2+p(1)^2+p(2)^2\\ \ge & 0 \end{align*}

Checkpoint 6.1.4.

Determine if the following functions meet the conditions of an inner product. Full proofs are not required.

(a)

\(\langle\vec{u},\vec{v}\rangle=u_1 v_1+u_2 v_2 \) for \(\R^n\) (real vectors of length \(n\text{.}\))

(b)

\(\langle p(x),q(x) \rangle = \int_0^2 p(x)q(x) \; dx \) for the space of integrable functions.

Subsection 6.1.3 Norm

For vectors in \(\R^n\) we talk about the magnitude (or norm) of the vector. In this section we use an inner product to define a generalized version of magnitude (norm).

Definition 6.1.5. Norm.

A function \(\|\vec{u}\|\) from a vector space \(U\) to \(\R\) is a norm if and only if

\(\|\vec{u}\| \ge 0\) for all \(\vec{u},\)
\(\|\vec{u}\|=0\) if and only if \(\vec{u}=\vec{0},\)
\(\|c\vec{u}\|=|c|\|\vec{u}\|,\) and
\(\|\vec{u}\|+\|\vec{v}\| \ge \|\vec{u}+\vec{v}\|\) for all \(\vec{u}, \vec{v}\text{.}\)

The last property is known as the triangle inequality. Use this activity to experiment with this concept.

Instructions.

Select lengths a,b,c for the line segments. Move the points to try to form a triangle.

Figure 6.1.6. Illustration of Triangle Inequality

Activity 6.1.1.

The triangle inequality’s simplest form is on a triangle with the standard Euclidean norm (distance formula). Here we use a triangle to experiment with this statement.

(a)

Set \(a=9\text{,}\) \(b=6\text{,}\) and \(c=4\text{.}\) Move points \(C\) and \(C^\prime\) so that a triangle is formed.

(b)

Set \(a=9.5\text{,}\) \(b=6\text{,}\) and \(c=4\text{.}\) Move points \(C\) and \(C^\prime\) so that a triangle is formed.

(c)

Set \(a=9.8\text{,}\) \(b=6\text{,}\) and \(c=4\text{.}\) Move points \(C\) and \(C^\prime\) so that a triangle is formed.

(d)

How do you have to move the points as side with length \(a\) gets longer?

(e)

Set \(a=11\text{,}\) \(b=6\text{,}\) and \(c=4\text{.}\) Try to move points \(C\) and \(C^\prime\) so that a triangle is formed. What happens?

In future courses we will study norms on their own. In this course we are interested in those norms that are created by an inner product.

Definition 6.1.7. Derived Norm.

The derived norm of an inner product is defined as

\begin{equation*} \|\vec{u}\| = \sqrt{\langle\vec{u},\vec{u}\rangle}. \end{equation*}

To become familiar with the derived norm we will illustrate the properties with examples.

Activity 6.1.2.

For these experiments use vectors from \(\R^2\) with the inner product

\begin{equation*} \langle\vec{u},\vec{v}\rangle = u_1 v_1+u_2 v_2+u_3 v_3 \end{equation*}

and its derived norm.

(a)

We illustrate that \(\|\vec{u}\| \ge 0\) for all \(\vec{u}\text{.}\)

(i)

Calculate \(\|[8,4,4]^T\|\)

(ii)

Calculate \(\|[-1,-8,5]^T\|\)

(iii)

Evaluate \(\|[v_1,v_2,v_3]^T\|\)

(iv)

Why must that always be non-negative?

(b)

We illustrate that \(\|\vec{u}\|=0\) if and only if \(\vec{u}=\vec{0}\text{.}\)

(i)

Calculate \(\|[0,0,0]^T\|\)

(ii)

Evaluate \(\|[v_1,v_2,v_3]^T\|\)

(iii)

Using the last step, if \(\|\vec{v}\|=0\text{,}\) why must \(\vec{v}=\vec{0}\text{?}\)

(c)

\(\|c\vec{u}\|=|c|\|\vec{u}\|\) for all \(c \in \R\) and \(\vec{u}\text{.}\)

(i)

Evaluate \(\|[8,4,4]^T\|\) and \(\|3[8,4,4]^T\|\text{.}\) Compare the results.

(ii)

Evaluate \(\|[-1,-8,5]^T\|\) and \(\|-2[-1,-8,5]^T\|\text{.}\) Compare the results.

Subsection 6.1.4 Orthogonal

Inner products are a generalization of the concept of dot product which has a connection to angles. Here we see how this also can be generalized.

Activity 6.1.3.

For these experiments use vectors from \(\R^2\) with the inner product

\begin{equation*} \langle\vec{u},\vec{v}\rangle = u_1 v_1+u_2 v_2. \end{equation*}

Note for this experiment larger and smaller are measured in absolute value.

(a)

If \(\vec{u}=(1,1)\text{,}\) what vectors \(\vec{v}\) will make \(\langle\vec{u},\vec{v}\rangle\) smallest? Remember smallest is closest to 0.

(b)

If \(\vec{u}=(1,1)\text{,}\) what vectors \(\vec{v}\) will make \(\langle\vec{u},\vec{v}\rangle\) largest? Remember largest means farthest from 0.

(c)

Describe the smallest and largest vector using geometric properties, that is, what direction do they point relative to \(\vec{u}=[1,1]^T\)

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