Range of a Linear Map Examples 1

Recall from the Range of a Linear Map page that if $T \in \mathcal L (V, W)$ then the range of $T$ denoted $\mathrm{range} (T)$ is the set of vectors $T(v)$ such that $v \in V$, that is:

(1)

\begin{align} \quad \mathrm{range} (T) = \{ T(v) : v \in V \} \end{align}

We noted that $\mathrm{range} (T) \neq \emptyset$ since $0 \in \mathrm{range} (T)$ and we also noted that $\mathrm{range} (T)$ was a subspace of the codomain vector space $W$.

We will now look at some problems regarding the range of a linear map.

Example 1

Let $T \in \mathcal L (V, W)$. Prove that if $V = \mathrm{span} (v_1, v_2, ..., v_n)$ then $\mathrm{range} (T) = \mathrm{span} (T(v_1), T(v_2), ..., T(v_n))$.

Let $V = \mathrm{span} (v_1, v_2, ..., v_n)$. Then any vector $v \in V$ can be written as a linear combination of the vectors in $\{ v_1, v_2, ..., v_n \}$ where $a_1, a_2, ..., a_n \in \mathbb{F}$ as:

(2)

\begin{align} \quad v = a_1v_1 + a_2v_2 + ... + a_nv_n \end{align}

Now apply the linear map $T$ to both sides of the equations above to get that:

(3)

\begin{align} \quad T(v) = T(a_1v_1 + a_2v_2 + ... + a_nv_n) \\ \quad T(v) = a_1T(v_1) + a_2T(v_2) + ... + a_nT(v_n) \\ \end{align}

Now since $\mathrm{range} (T) = \{ T(v) : v \in V$ then from the equation above we have that:

(4)

\begin{align} \quad \mathrm{range} (T) = \{ a_1T(v_1) + a_2T(v_2) + ... + a_nT(v_n) : a_1, a_2, ..., a_n \in \mathbb{F} \} = \mathrm{span} (T(v_1), T(v_2), ..., T(v_n)) \end{align}

Therefore $\mathrm{range} (T) = \mathrm{span} (T(v_1), T(v_2), ..., T(v_n))$.