# 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)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)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)We have that:

(4)Thus we have that:

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

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

(7)