Let L:V→W be a linear transformation. Explain the meaning of the following statement: The action...

Step 1
Given:
${\displaystyle L:V\to W}$ is a linear transformation.
Now, if ${\displaystyle \{v_1,v_2,\dots ,v_n\}}$ is a basis for V such that:
${\displaystyle L\left(v_i\right)=w_i}$, for all i,
Then, the action of L on any element of V can be determined.
Step 2
Let v be in V.
Then,
${\displaystyle v=a_1{v}_1+a_{2}L\left(v_2\right)+\dots +a_n{v}_n}$
So,
${\displaystyle L\left(v\right)=a_{1}L\left(v1\right)+a_{2}L\left(v_2\right)+\dots +a_{n}L\left(v_n\right)}$
${\displaystyle L\left(v\right)=a_1{w}_1+a_2{w}_2+\dots .+a_n{w}_n}$
Therefore, L(v) is determined.
Thus, “the action of L is completely determined by its action on a basis for V”.