Linear Functionals Examples 1

# Linear Functionals Examples 1

Recall from the Linear Functionals page that if $V$ is a vector space over the field $\mathbb{F}$ then a linear functional $\varphi : V \to \mathbb{F}$ is a linear map from $V$ to the field $\mathbb{F}$.

One of the important theorems that we saw is that if $V$ is a finite-dimensional inner product space and $\varphi$ is a linear function on $V$ then there exists a unique vector $v \in V$ such that for every $u \in V$ we have that:

(1)
\begin{align} \quad \varphi (u) = <u, v> \end{align}

Provided that $\{ e_1, e_2, ..., e_n \}$ is an orthonormal basis of $V$, we have that then this unique vector $v \in V$ is given by:

(2)
\begin{align} \quad v = \overline{\varphi(e_1)}e_1 + \overline{\varphi(e_2)}e_2 + ... + \overline{\varphi(e_n)}e_n \end{align}

We will now look at some examples regarding linear functionals.

## Example 1

Consider the inner product space $\mathbb{R}^3$ where the inner product defined on $\mathbb{R}^3$ is the dot product, $<(x_1, x_2, x_3), (y_1, y_2, y_3)> = x_1y_1 + x_2y_2 + x_3y_3$. Define a linear functional by $\varphi (x_1, x_2, x_3) = x_1 + 2x_2 + 3x_3$. Find the unique vector in $(z_1, z_2, z_3) \in \mathbb{R}^3$ such that $\varphi (x_1, x_2, x_3) = <(x_1, x_2, x_3), (z_1, z_2, z_3)>$ for all $(x_1, x_2, x_3) \in \mathbb{R}^3$.

Let the standard basis $\{ e_1, e_2, e_3 \} = \{ (1, 0, 0), (0, 1, 0), (0, 0, 1) \}$ be our orthonormal basis. Note that this basis is orthonormal WITH RESPECT TO our inner product defined to be the dot product (this is where the inner product plays a role in determining $(z_1, z_2, z_3)$). Since we're working over the real numbers, we have that:

(3)
\begin{align} \quad (z_1, z_2, z_3) = \varphi (e_1) e_1 + \varphi (e_2) e_2 + \varphi (e_3)e_3 \end{align}

We have that:

(4)
\begin{align} \quad \varphi (e_1) = \varphi (1, 0, 0) = 1 \\ \quad \varphi (e_2) = \varphi (0, 1, 0) = 2 \\ \quad \varphi (e_3) = \varphi (0, 0, 1) = 3 \end{align}

Thus we have that:

(5)
\begin{align} \quad (z_1, z_2, z_3) = 1(1, 0, 0) + 2(0, 1, 0) + 3(0, 0, 1) \\ \quad (z_1, z_2, z_3) = (1, 2, 3) \end{align}

Therefore, for every $(x_1, x_2, x_3) \in \mathbb{R}^3$ we have that:

(6)
\begin{align} \quad \varphi (x_1, x_2, x_3) = <(x_1, x_2, x_3), (1, 2, 3)> \end{align}

To verify this, just expand the inner product on the righthand side to get that:

(7)
\begin{align} \quad <(x_1, x_2, x_3), (1, 2, 3)> = x_1 + 2x_2 + 3x_3 = \varphi(x_1, x_2, x_3) \end{align}