Image and Spans Theorem

Theorem

• if $\{v_1,\dots ,v_n\}$ is a Spanning Set for $V$
• If $T:V\to W$ is Linear Transformation
• then, $\{T(v_1),\dots ,T(v_2)\}$ is a spanning set for $Image(T)$

Proof

Proving $span\{T(v_1),\dots ,T(v_n)\}\subset Image(T)$

1. Pick $w\in span\{T(v_1),\dots ,T(v_n)\}$
2. Then, $w=a_{1}T(v_1)+\cdots +a_{n}T(v_n)$
3. $=T(a_1{v}_1+\cdots +a_n{v}_n)$ by Linearity
4. $\in Image(T)$

Proving $Image(T)\subset span\{T(v_1),\dots ,T(v_n)\}$

1. Pick $\overrightarrow{w}\in Image(T)$
2. We can express $\overrightarrow{w}=T(\overrightarrow{v})$
3. $=T(a_1{v}_1+\dots a_n{v}_n)$ as $\{v_1,\dots ,v_n\}$ is a spanning set
4. $=a_{1}T(v_1)+\cdots +a_{n}T(v_n)$ by Linearity
5. $\in span\{T(v_1),\dots ,T(v_n)\}$ by defn of span