Direct Sum

3. Linear transformation between direct sums

Let V = V1V2 and W = W1W2 be direct sums. We express the vectors in V and W vertically

 v = [ v1 ] ∈ V, w = [ w1 ] ∈ W, where v1 ∈ V1, v2 ∈ V2, w1 ∈ W1, w2 ∈ W2. v2 w2

Then a linear transformation T: VW is of the form

 T(v) = [ T1(v) ], where T1: V → W1 and T2: V → W2. T2(v)

From

 T(u + v) = [ T1(u + v) ], T(u) + T(v) = [ T1(u) + T1(v) ], T2(u + v) T2(u) + T2(v)

we get T1(u + v) = T1(u) + T1(v), T2(u + v) = T2(u) + T2(v). Similarly, we have T1(cu) = cT1(u), T2(cu) = cT2(u). Thus T1 and T2 are linear.

The linearity of T1 implies T1(v) = T11(v1) + T12(v2), where

 T11(v1) = T1[ v1 ]: V1 → W1, T12(v2) = T1[ 0 ]: V2 → W1 0 v2

are also linear transformations. Similarly, we have T2(v) = T21(v1) + T22(v2) for linear transformations T21: V1W2 and T22: V2W2. In summary, we have

 T(v) = [ T1(v) ] = [ T11(v1) + T12(v2) ] = [ T11 T12 ] [ v1 ]. T2(v) T21(v1) + T22(v2) T21 T22 v2

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: V1V2⊕...⊕VkW1W2⊕...⊕Wl

is given by

 T[ v1 ] = [ T11 T12 . . T1k ] [ v1 ], v2 T21 T22 . . T2k v2 : : : : : vk Tl1 Tl2 . . Tlk vk

where Tij: VjWi are linear transformations.