Direct sum of vector spaces

Direct sum of linear maps

For linear endomorphisms 𝐴 :𝑉 →𝑉 and 𝐵 :𝑊 →𝑊, the direct sum is defined as follows

𝐴⊕𝐵:𝑉⊕𝑊→𝑉⊕𝑊(𝑣,𝑤)↦(𝐴𝑣,𝐴𝑤)

This is used to define the Direct sum of representations

Matrices

If 𝐀 and 𝐁 are matrices, then 𝐀 ⊕𝐁 is the Block matrix expanded from

[𝐀00𝐁][⃗𝐯⃗𝐰]=[𝐀⃗𝐯𝐁⃗𝐰]

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