Discover Linear Algebra

Example 36.5.1. An inner product on $\poly_n(\R)$.

In Discovery 36.2, we verified the four real inner product axioms for an example inner product on $\poly_2(\R)\text{,}$ the space of polynomials with real coefficients of degree $2$ or less. We can mimic this example to create an inner product on $\poly_n(\R)$ for any $n\text{:}$ choose $n+1$ distinct real numbers $c_0, c_1, \dotsc, c_n\text{,}$ and using them create pairing

Checking Axiom RIP 1, Axiom RIP 2, and Axiom RIP 3 is straightforward. And The Fundamental Theorem of Algebra (Real Version) guarantees that no nonzero polynomial $p$ can evaluate to zero at more than $n$ input values, hence not all of the terms in the pairing expression

can be zero, which verifies Axiom RIP 4.

Example 36.5.2. The standard inner product on $\matrixring_{m\times n}(\R)$.

An $m \times n$ matrix is just $mn$ “components” (i.e. entries) arranged in a grid instead of in a column. So we would expect the pairing

which is really just a “dot product” of matrices, to create an inner product on $\matrixring_{m\times n}(\R)\text{.}$ And it does.

We can wrap this pairing up in a neat formula by

(Again, the reversal of order is in preparation of the complex version.)

Let's verify Axiom RIP 1:

with justifications

1. definition of the pairing;
2. Rule 5.a of Proposition 4.5.1;
3. Rule 5.d of Proposition 4.5.1;
4. transpose does not change the diagonal entries, so trace remains the same; and
5. definition of the pairing.

Axiom RIP 2 and Axiom RIP 3 are also easily verified using the properties of transpose and trace. So let's finish this example by verifying Axiom RIP 4. Consider a matrix $A$ as being made up of column vectors in $\R^n\text{:}$

Then the diagonal entries of $\utrans{A} A$ are of the form

If $A \neq \zerovec\text{,}$ then at least one of its columns $\uvec{a}_j$ must be nonzero, and that column will contribute the positive value $\norm{\uvec{a}_j}^2$ to

Example 36.5.3. The standard inner product on $\matrixring_{m\times n}(\C)$.

Similar to the real case, we can effectively make a complex matrix “dot product” by setting

Again, we can achieve this result with the compact formula

We leave it to you, the reader, to verify that this pairing will satisfy the axioms for a complex inner product.

Example 36.5.4. An inner product for continuous functions.

Let $C[a,b]$ represent the space of all continuous functions on the closed interval $a \le x \le b\text{.}$ Since adding continuous functions or vertically scaling a continuous function always results in a continuous function, this is a subspace of $F[a,b]\text{,}$ the space of all functions defined on domain $a \le x \le b\text{.}$

Define a pairing on $C[a,b]$ by

A product of two continuous functions is also continuous, and the Fundamental Theorem of Calculus tells us that continuous functions are always integrable.

This pairing obviously satisfies Axiom RIP 1, and the basic properties of definite integrals tell us that Axiom RIP 2 and Axiom RIP 3 are also satisfied. For Axiom RIP 4, consider that

must at least be nonnegative because the integrand is, but if $f(x)$ is not the zero function, then the properties of continuous functions require that this integral will evaluate to a positive number.

Example 36.5.5. The “length” of a matrix.

Let's use the inner product

on $\matrixring_{2 \times 2}(\R)$ to compute the norm of the vector

We have

and so

What unit vectors in $\matrixring_{2 \times 2}(\R)$ are “parallel” to $A\text{?}$ Just as in $\R^n\text{,}$ we can normalize a vector to a unit vector by dividing by its norm. So

is one unit vector that is “parallel” to $A\text{,}$ and $-U$ is another.

Example 36.5.6. Angle between matrices.

What is the angle between

in $\matrixring_{2 \times 2}(\R)$ when using the inner product $\inprod{X}{Y} = \trace (\utrans{B} A)\text{?}$

Compute:

Put these calculations together in the formula

Example 36.5.7. Orthogonal functions.

The functions $f(x) = \sin x$ and $g(x) = \cos x$ are continuous, and so are vectors in $C[0,2\pi]\text{.}$ If we use the inner product of Example 36.5.4 to compute

we find that the angle between these functions is

so that $f$ and $g$ are at a right angle to each other.