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Discovery guide 36.2 Discovery guide

A pairing of vectors is called a (real) inner product if it satisfies the four axioms below.

List 36.2.1. (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 \(\uvecinprod{v}{v} \gt 0\text{.}\)

Discovery 36.1.

Convince yourself that the pairing \(\uvecinprod{v}{w} = \udotprod{v}{w}\) satisfies the inner product axioms.

Hint

Proposition 13.5.3.

Discovery 36.2.

Consider the pairing

\begin{equation*} \inprod{p}{q} = p(-1)q(-1) + p(0)q(0) + p(1)q(1) \end{equation*}

on \(\poly_2(\R)\text{,}\) the space of all polynomials of degree \(2\) or less.

(a)

For each of Axiom RIP 1, Axiom RIP 2, and Axiom RIP 3, rewrite the left- and right-hand sides of the axiom equality using the above definition of the pairing.

By comparing left- and right-hand sides, convince yourself that the pairing will satisfy each of those axioms.

(c)

Would Axiom RIP 4 hold if we used this pairing on \(\poly_3(\R)\text{,}\) the space of all polynomials of degree \(3\) or less?

If not, how would you “fix” it?

Discovery 36.3.

Combining Axiom RIP 2, and Axiom RIP 3, we can say that an inner product is linear in the first term:

\begin{equation*} \inprod{a \uvec{u} + b \uvec{v}}{\uvec{w}} = a \uvecinprod{u}{w} + b \uvecinprod{v}{w} \text{.} \end{equation*}

Must an inner product also be linear in the second term?

Discovery 36.4.

Use Axiom RIP 3 for a special choice of \(k\) to determine that an inner product must satisfy

\begin{equation*} \inprod{\zerovec}{\uvec{v}} = 0 \end{equation*}

for every vector \(\uvec{v}\) in the vector space.

What about \(\inprod{\uvec{v}}{\zerovec} \text{?}\)

Discovery 36.5.

Recall that eventually we would like the concept of inner product to allow us to do geometry in vector spaces besides \(\R^n\text{.}\) As discussed in Section 36.1, the connection between the dot product and geometry in \(\R^n\) is

\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*}

Based on this, what is the point of including Axiom RIP 4 in our list of axioms?

Discovery 36.6.
(a)

Compute the “dot product” \(\udotprod{w}{w}\) for the \(\C^2\) vector \(\uvec{w} = (\ci,\ci)\text{.}\) Is this going to work out for us, given the geometric reason for including Axiom RIP 4 in our list axioms that we explored in Discovery 36.5?

(b)

Let's back up a dimension. We already know how to compute length in \(\C^1\text{,}\) by regarding complex numbers as vectors in the complex plane. Use this point of view to propose a pairing on \(\C^1\) that will satisfy Axiom RIP 4. (Make sure to define your pairing as a pairing of two possibly different \(\C^1\) vectors.)

Hint

Rule 1 in Proposition A.2.14.

(c)

Use the pattern of your pairing for \(\C^1\) from Task b to propose a pairing on \(\C^2\) that will satisfy Axiom RIP 4. (Again, make sure to define your pairing as a pairing of two possibly different \(\C^2\) vectors.)

(d)

Which of the other three inner product axioms does your proposed pairing for \(\C^2\) from Task c satisfy?

Will your pairing still be linear in both terms? (See Discovery 36.3.)

The dot product is not the only inner product on \(\R^n\text{.}\)

Discovery 36.7.

Recall that the dot product on \(\R^n\) can be realized using matrix multiplication:

\begin{equation*} \udotprod{v}{w} = \utrans{\uvec{w}} \uvec{v} \text{.} \end{equation*}

(See Subsection 13.3.9.)

If we want to generalize in order to look for other, similar inner products, the first step is to look for an underlying pattern that can be changed. Notice that we can sneak an identity matrix in there:

\begin{equation*} \udotprod{v}{w} = \utrans{\uvec{w}} I \uvec{v} \text{.} \end{equation*}

What if we change \(I\) to some other matrix \(A\text{?}\) Is

\begin{gather} \uvecinprod{v}{w} = \utrans{\uvec{w}} A \uvec{v}\label{equation-inner-prod-discovery-modified-euclidean}\tag{\(\star\)} \end{gather}

still an inner product?

Let's investigate using vectors in \(\R^2\) and a \(2 \times 2\) matrix

\begin{equation*} A = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} \text{.} \end{equation*}

Recall that \(\uvec{e}_1,\uvec{e}_2\) represent the standard basis vectors in \(\R^2\text{.}\)

(a)

What does using \(\uvec{e}_1,\uvec{e}_2\) in Axiom RIP 1 tell you about \(b\) and \(c\text{,}\) if the axiom is going to be true for pairing (\(\star\))? What does this say about \(A\text{?}\)

(b)

What does using \(\uvec{e}_1\) and \(\uvec{e}_2\) (separately) in Axiom RIP 4 say about \(a\) and \(d\text{,}\) if the axiom is going to be true for pairing (\(\star\))?

(c)

Calculate \(\uvecinprod{x}{x}\) for \(\uvec{x} = (d,-b)\text{.}\) After simplifying, you should find \(\det A\) hiding in the result.

If Axiom RIP 4 is to be true, what does \(\uvecinprod{x}{x}\) for this \(\uvec{x}\) tell you about \(\det A\text{?}\)