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)

\begin{align} \quad T(x + y) = T((x_1, y_1, z_1) + (x_2, y_2, z_2)) = T(x_1 + x_2, y_1 + y_2, z_1 + z_2) \\ = ([x_1 + x_2] + [y_1 + y_2], [x_1 + x_2] + 2[z_1 + z_2], [y_1 + y_2] - [z_1 + z_2]) \\ = (x_1 +y_1, x_1 + 2z_1, y_1 - z_1) + (x_2 + y_2, x_2 + 2z_2, y_2 - z_2) = T(x) + T(y) \end{align}

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)

\begin{align} \quad T(ax) = T(a(x_1, y_1, z_1)) = T(ax_1, ay_1, az_1) = (ax_1 + ay_1, ax_1 + 2az_1, ya_1 - za_1) \\ \quad =a(x_1 + y_1, x_1 + 2z_1, y_1 - z_1) = aT(x) \end{align}

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)

\begin{align} \quad T(x + y) = T((x_1, y_1) + (x_2, y_2)) = T(x_1 + x_2, y_1 + y_2) \\ = ([x_1 + x_2] + b[x_1 + x_2][y_1 + y_2], 2[x_1 + x_2] + 3[y_1 + y_2], [x_1 + x_2] - c \cos [x_1 + x_2] ) \end{align}

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

(4)

\begin{align} \quad T(x) + T(y) = T(x_1, y_1) + T(x_2, y_2) = (x_1 + bx_1y_1, 2x_1 + 3y_1, x_1 - c \cos (x_1)) + (x_2 + bx_2y_2, 2x_2 + 3y_2, x_2 - c \cos (x_2)) \end{align}

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

(5)

\begin{align} \quad b[x_1 + x_2][y_1 + y_2] = b(x_1y_1 + x_2y_2) \quad \mathrm{and} \quad c \cos [x_1 + x_2] = c (\cos (x_1) + \cos (x_2)) \end{align}

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)

\begin{align} \quad T(ax) = T(a(x_1, y_1)) = T(ax_1, ay_1) = (ax_1 + ba^2x_1y_1, 2ax_1 + 3ay_1, ax_1 - c \cos (ax_1)) \end{align}

We also have that:

(7)

\begin{align} \quad aT(x) = aT(x_1, y_1) = a(x_1 + bx_1y_1, 2x_1 + 3y_1, x_1 - c \cos (x_1)) = (ax_1 + bax_1y_1, 2ax_1 + 3ay_1, ax_1 - ca \cos (x_1)) \end{align}

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

(8)

\begin{align} \quad bax_1y_1 = ba^2x_1y_1 \quad \mathrm{and} \quad c \cos (ax_1) = ca \cos (x_1) \end{align}

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.