Let V = V_{1}⊕V_{2} and W = W_{1}⊕W_{2} be direct sums. We express the vectors in V and W vertically
v = [ | v_{1} | ] ∈ V, w = [ | w_{1} | ] ∈ W, where v_{1} ∈ V_{1}, v_{2} ∈ V_{2}, w_{1} ∈ W_{1}, w_{2} ∈ W_{2}. |
v_{2} | w_{2} |
Then a linear transformation T: V → W is of the form
T(v) = [ | T_{1}(v) | ], where T_{1}: V → W_{1} and T_{2}: V → W_{2}. |
T_{2}(v) |
From
T(u + v) = [ | T_{1}(u + v) | ], T(u) + T(v) = [ | T_{1}(u) + T_{1}(v) | ], |
T_{2}(u + v) | T_{2}(u) + T_{2}(v) |
we get T_{1}(u + v) = T_{1}(u) + T_{1}(v), T_{2}(u + v) = T_{2}(u) + T_{2}(v). Similarly, we have T_{1}(cu) = cT_{1}(u), T_{2}(cu) = cT_{2}(u). Thus T_{1} and T_{2} are linear.
The linearity of T_{1} implies T_{1}(v) = T_{11}(v_{1}) + T_{12}(v_{2}), where
T_{11}(v_{1}) = T_{1}[ | v_{1} | ]: V_{1} → W_{1}, T_{12}(v_{2}) = T_{1}[ | 0 | ]: V_{2} → W_{1} |
0 | v_{2} |
are also linear transformations. Similarly, we have T_{2}(v) = T_{21}(v_{1}) + T_{22}(v_{2}) for linear transformations T_{21}: V_{1} → W_{2} and T_{22}: V_{2} → W_{2}. In summary, we have
T(v) = [ | T_{1}(v) | ] = [ | T_{11}(v_{1}) + T_{12}(v_{2}) | ] = [ | T_{11} | T_{12} | ] [ | v_{1} | ]. |
T_{2}(v) | T_{21}(v_{1}) + T_{22}(v_{2}) | T_{21} | T_{22} | v_{2} |
In the last equality, we consider T(v) as product of T and v.
The computation illustrates that linear transformations between direct sums are given by "partitioned transformations", comparable to partitioned matrix.
By writing vectors in direct sums vertically and thinking of T(v) as product of T and v, a linear transformation
T: V_{1}⊕V_{2}⊕...⊕V_{k} → W_{1}⊕W_{2}⊕...⊕W_{l}
is given by
T[ | v_{1} | ] = [ | T_{11} | T_{12} | . . | T_{1k} | ] [ | v_{1} | ], |
v_{2} | T_{21} | T_{22} | . . | T_{2k} | v_{2} | ||||
: | : | : | : | : | |||||
v_{k} | T_{l1} | T_{l2} | . . | T_{lk} | v_{k} |
where T_{ij}: V_{j} → W_{i} are linear transformations.