Linear Maps Examples 4

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

Let $V$ be a vector space and let $U$ be a subspace of $V$ such that $U \neq V$. Let $S \in \mathcal L (U, W)$ where $S$ is not the zero map, and define a function $T : V \to W$ by $T(v) = \left\{\begin{matrix} S(v) \quad \mathrm{if} \: v \in U \\ 0 \quad \mathrm{if} \: v \not \in U \end{matrix}\right.$. Prove that $T$ is NOT a linear map.

To show that $T$ is not a linear map, we must show that either the additivity or the homogeneity property does not hold for some vectors/scalars in $V$/$\mathbb{F}$.

Choose a vector $v \in U$ such that $S(v) \neq 0$. This is possible since $S$ is not identically equal to the zero map. Also choose a vector $w \not \in U$. Then we have that $(v + w) \not \in U$ (since if $(v + w) \in U$ then since $v \in U$ we have by the closure under addition of $U$ that $(v + w) - (v) = w \in U$ which is a contradiction). Therefore since $(v + w) \not \in U$ we have that:

(1)

\begin{align} \quad T(v + w) = 0 \end{align}

However, we note that:

(2)

\begin{align} \quad T(v) + T(w) = S(v) + 0 = S(v) \neq 0 \end{align}

Therefore $T(v + w) = 0 \neq T(v) + T(w)$ and so the additivity property does not hold, and so $T$ is not a linear map.