A function \(L \colon V\rightarrow W\) is linear if \(V\) and \(W\) are vector spaces and for all \(u,v \in V\) and \(r,s \in \Re\) we have
\[ L(ru + sv) = rL(u) + sL(v) .\]
We will often refer to linear functions by names like "linear map'', "linear operator'' or "linear transformation''. In some contexts you will also see the name "homomorphism''. The definition above coincides with the two part description in chapter 1; the case \(r=1,s=1\) describes additivity, while \(s=0\) describes homogeneity. We are now ready to learn the powerful consequences of linearity.
- 6.3: Linear Differential Operators
- Your calculus class became much easier when you stopped using the limit definition of the derivative, learned the power rule, and started using linearity of the derivative operator.
Thumbnail: A linear combination of one basis set of vectors (purple) obtains new vectors (red). If they are linearly independent, these form a new basis set. The linear combinations relating the first set to the other extend to a linear transformation, called the change of basis. (CC0; Maschen via Wikipedia)