Extension Theorem

linear-algebra

Extension Theorem

Let $V$ be a vector space and $B$ be the set of basis vectors of $V$. Let $W$ be a vector spaces and $f:V\to W$ be a linear mapping.

$$\begin{array}{rl}\mathbf{x}& ={\mathbf{x}}_1\cdot {\mathbf{b}}_1+\dots +{\mathbf{x}}_n\cdot {\mathbf{b}}_n⟹\\ f(\mathbf{x})& ={\mathbf{x}}_1\cdot f({\mathbf{b}}_1)+\dots +{\mathbf{x}}_n\cdot f({\mathbf{b}}_n)\end{array}$$

where $f({\mathbf{b}}_i)$ is called the image of the basis vector ${\mathbf{b}}_i$.

Conversely, this formula always defines a linear mapping $f:V\to W$ for any choice of $f({\mathbf{b}}_j)\in W$. Thus, it is enough to know the images of the basis vectors to characterise a linear mapping $f:V\to W$.