It is the interaction between linear transformations and linear combinations that lies at the heart of many of the important theorems of linear algebra. The next theorem distills the essence of this. The proof is not deep, the result is hardly startling, but it will be referenced frequently. We have already passed by one occasion to employ it, in the proof of Theorem MLTCV. Paraphrasing, this theorem says that we can “push” linear transformations “down into” linear combinations, or “pull” linear transformations “up out” of linear combinations. We will have opportunities to both push and pull.
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation, $\vectorlist{u}{t}$ are vectors from $U$ and $\scalarlist{a}{t}$ are scalars from $\complex{\null}$. Then \begin{equation*} \lt{T}{\lincombo{a}{u}{t}} = a_1\lt{T}{\vect{u}_1}+ a_2\lt{T}{\vect{u}_2}+ a_3\lt{T}{\vect{u}_3}+\cdots+ a_t\lt{T}{\vect{u}_t} \end{equation*}
Some authors, especially in more advanced texts, take the conclusion of Theorem LTLC as the defining condition of a linear transformation. This has the appeal of being a single condition, rather than the two-part condition of Definition LT. (See Exercise LT.T20).
Our next theorem says, informally, that it is enough to know how a linear transformation behaves for inputs from any basis of the domain, and all the other outputs are described by a linear combination of these few values. Again, the statement of the theorem, and its proof, are not remarkable, but the insight that goes along with it is very fundamental.
Suppose $U$ is a vector space with basis $B=\set{\vectorlist{u}{n}}$ and the vector space $V$ contains the vectors $\vectorlist{v}{n}$ (which may not be distinct). Then there is a unique linear transformation, $\ltdefn{T}{U}{V}$, such that $\lt{T}{\vect{u}_i}=\vect{v}_i$, $1\leq i\leq n$.
You might recall facts from analytic geometry, such as “any two points determine a line” and “any three non-collinear points determine a parabola.” Theorem LTDB has much of the same feel. By specifying the $n$ outputs for inputs from a basis, an entire linear transformation is determined. The analogy is not perfect, but the style of these facts are not very dissimilar from Theorem LTDB.
Notice that the statement of Theorem LTDB asserts the existence of a linear transformation with certain properties, while the proof shows us exactly how to define the desired linear transformation. The next two examples show how to compute values of linear transformations that we create this way.
Here is a third example of a linear transformation defined by its action on a basis, only with more abstract vector spaces involved.
Informally, we can describe Theorem LTDB by saying “it is enough to know what a linear transformation does to a basis (of the domain).”