Direct sum of linear maps

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

𝐴 \oplus 𝐵 : 𝑉 \oplus 𝑊 \to 𝑉 \oplus 𝑊 (𝑣, 𝑤) \mapsto (𝐴 𝑣, 𝐴 𝑤)

Matrices

If $𝐀$ and $𝐁$ are matrices, then $𝐀 \oplus 𝐁$ is the Block matrix expanded from

[𝐀 0 0 𝐁] [⃗ 𝐯 ⃗ 𝐰] = [𝐀 ⃗ 𝐯 𝐁 ⃗ 𝐰]

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