We can combine linear transformations in natural ways to create new linear transformations. So we will define these combinations and then prove that the results really are still linear transformations. First the sum of two linear transformations.
Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{U}{V}$ are two linear transformations with the same domain and codomain. Then their sum is the function $\ltdefn{T+S}{U}{V}$ whose outputs are defined by \begin{equation*} \lt{(T+S)}{\vect{u}}=\lt{T}{\vect{u}}+\lt{S}{\vect{u}} \end{equation*}
Notice that the first plus sign in the definition is the operation being defined, while the second one is the vector addition in $V$. (Vector addition in $U$ will appear just now in the proof that $T+S$ is a linear transformation.) Definition LTA only provides a function. It would be nice to know that when the constituents ($T$, $S$) are linear transformations, then so too is $T+S$.
Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{U}{V}$ are two linear transformations with the same domain and codomain. Then $\ltdefn{T+S}{U}{V}$ is a linear transformation.
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation and $\alpha\in\complex{\null}$. Then the scalar multiple is the function $\ltdefn{\alpha T}{U}{V}$ whose outputs are defined by \begin{equation*} \lt{(\alpha T)}{\vect{u}}=\alpha\lt{T}{\vect{u}} \end{equation*}
Given that $T$ is a linear transformation, it would be nice to know that $\alpha T$ is also a linear transformation.
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation and $\alpha\in\complex{\null}$. Then $\ltdefn{(\alpha T)}{U}{V}$ is a linear transformation.
Now, let us imagine we have two vector spaces, $U$ and $V$, and we collect every possible linear transformation from $U$ to $V$ into one big set, and call it $\vslt{U}{V}$. Definition LTA and Definition LTSM tell us how we can “add” and “scalar multiply” two elements of $\vslt{U}{V}$. Theorem SLTLT and Theorem MLTLT tell us that if we do these operations, then the resulting functions are linear transformations that are also in $\vslt{U}{V}$. Hmmmm, sounds like a vector space to me! A set of objects, an addition and a scalar multiplication. Why not?
Suppose that $U$ and $V$ are vector spaces. Then the set of all linear transformations from $U$ to $V$, $\vslt{U}{V}$, is a vector space when the operations are those given in Definition LTA and Definition LTSM.
Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{V}{W}$ are linear transformations. Then the composition of $S$ and $T$ is the function $\ltdefn{(\compose{S}{T})}{U}{W}$ whose outputs are defined by \begin{equation*} \lt{(\compose{S}{T})}{\vect{u}}=\lt{S}{\lt{T}{\vect{u}}} \end{equation*}
Given that $T$ and $S$ are linear transformations, it would be nice to know that $\compose{S}{T}$ is also a linear transformation.
Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{V}{W}$ are linear transformations. Then $\ltdefn{(\compose{S}{T})}{U}{W}$ is a linear transformation.
Here is an interesting exercise that will presage an important result later. In Example STLT compute (via Theorem MLTCV) the matrix of $T$, $S$ and $T+S$. Do you see a relationship between these three matrices?
In Example SMLT compute (via Theorem MLTCV) the matrix of $T$ and $2T$. Do you see a relationship between these two matrices?
Here is the tough one. In Example CTLT compute (via Theorem MLTCV) the matrix of $T$, $S$ and $\compose{S}{T}$. Do you see a relationship between these three matrices???