Direct sum vector space

The direct sum of vector spaces is the coproduct of vector spaces. #m/def/linalg It may be constructed as tuples with componentwise operations (cf. Direct sum of modules).

Internal direct sum

Let $𝑉$ be a vector space and ${𝑆 𝑖} 𝑖 \in 𝐼$ be a family of subspaces. Then $𝑉$ is the direct sum $⨁ 𝑖 \in 𝐼 𝑆 𝑖$ iff $𝑉 = \sum 𝑖 \in 𝐼 𝑆 𝑖$ and #m/def/linalg

𝑆 𝑖 \cap ⎛ ⎜ ⎜ ⎝ \sum 𝑗 \neq 𝑖 𝑆 𝑖 ⎞ ⎟ ⎟ ⎠ = {0}

If $𝑆 1 ⨁ 𝑆 2 = 𝑉$ , then $𝑆 2$ is a complement of $𝑆 1$ .¹

Further characterisations

Fixed basis

Let $𝑉, 𝑊 \in 𝖵 𝖾 𝖼 𝗍 𝕂$ be vector spaces over $𝕂$ with bases ${𝑣 𝑖} 𝑛 𝑖 = 1$ and ${𝑤 𝑗} 𝑚 𝑗 = 1$ respectively. The direct sum $𝑉 \oplus 𝑊$ of these spaces then has basis ${𝑣 𝑖} 𝑛 𝑖 = 1 ⨿ {𝑤 𝑗} 𝑚 𝑗 = 1$ .

Inner product spaces

If $𝑉$ and $𝑊$ are inner product spaces, then $⟨ (𝑣 1, 𝑤 1) | (𝑣 2, 𝑤 2) ⟩ = ⟨ 𝑣 1 | 𝑣 2 ⟩ + ⟨ 𝑤 1 | 𝑤 2 ⟩$

Properties

$d i m (𝑉 \oplus 𝑊) = d i m 𝑉 + d i m 𝑊$