# Linear Maps Examples 2

Recall from the Linear Maps page that a linear map or linear transformation from the vector space $V$ to the vector space $W$ is a function $T : V \to W$ such that for all $u, v \in V$ and for all $a \in \mathbb{F}$ we have that $T(u + v) = T(u) + T(v)$ (additivity property) and $T(av) = aT(v)$ (homogeneity property).

We will now look at some more example questions regarding linear maps.

## Example 1

**Determine whether the function $T : \mathbb{R}^3 \to \mathbb{R}^3$ defined by $T(x, y, z) = (x + y, x + 2z, y - z)$ for all $x = (x, y, z) \in \mathbb{R}^3$ is a linear map.**

We will first show that the additivity property holds. Let $x = (x_1, y_1, z_1), y = (x_2, y_2, z_2) \in \mathbb{R}^3$. Then we have that:

(1)Therefore the additivity property holds. We will now show that the homogeneity property holds. Let $x = (x_1, y_1, z_1) \in \mathbb{R}^3$ and let $a \in \mathbb{F}$. Then:

(2)Therefore the homogeneity property holds. Therefore $T$ is indeed a linear map from $\mathbb{R}^3$ to $\mathbb{R}^3$.

## Example 2

**Consider the linear map $T : \mathbb{R}^2 \to \mathbb{R}^3$ defined by $T(x, y) = (x + bxy, 2x + 3y, x - c \cos (x))$. Determine for what values of $b$ and $c$ make $T$ a linear map from $\mathbb{R}^2$ to $\mathbb{R}^3$.**

Let's first check the additivity property. Let $x = (x_1, y_1), y = (x_2, y_2) \in \mathbb{R}^2$. Then we have that:

(3)Now let's compute $T(x) + T(y)$ and compare both sides. We have that:

(4)In comparing $T(x + y)$ and $T(x) + T(y)$ we see that we must have that:

(5)Neither of these expressions equal each other for all values of $x_1, x_2, y_1, y_2 \in \mathbb{R}$, and so we must have that $b = 0$ and $c = 0$ for the additivity property to hold.

Now let's look at the homogeneity property. Let $x = (x_1, y_1) \in \mathbb{R}^2$ and let $a \in \mathbb{F}$. Then we have that:

(6)We also have that:

(7)From the equations for $T(ax)$ and $aT(x)$ we see that $T(ax) = aT(x)$ if and only if:

(8)Once again, we see that these equations hold for all $x_1, y_1 \in \mathbb{R}$ and for all $a \in \mathbb{F}$ if and only if $b = 0$ and $c = 0$.

Thus if $b = c = 0$, then $T$ is a linear map from $\mathbb{R}^2$ to $\mathbb{R}^3$, and otherwise, $T$ is not.