Linear Mapping Between Two Ordered Basis Theorem

Theorem

• For Vector Spaces $V,W$
• With set $\{a_1,\dots ,a_n\}$ as Ordered Basis for $V$
• With set $\{b_1,\dots ,b_n\}$ as Ordered Basis for $W$
• There exists a $T:V\to W$ s.t $T(a_i)=b_i,\forall i\in [1,n]$

Intuition

If we know what $T$ does to a basis, then we know $T$ completely