5.11 Direct sums

Our treatment of vector spaces has alternated between the fairly concrete examples of R² and R³ and the more abstract definition presented in section 5.2. I continue going back and forth with the fundamental idea of the direct sum of two vector spaces over the same field F. direct sum vector space$direct sum ⊕ (direct sum)

In a vague and neither concrete nor abstract sense, a direct sum is when you push two vector spaces together. It’s one of the ways you can construct a new vector space from existing ones.

Let V and W be two vector spaces of dimensions n and m over F. If we write v = (v₁, v₂, …, v_n) and w = (w₁, w₂, …, w_m), then

in the direct sum vector space V ⊕ W. It has dimension n + m.

All the requirements regarding addition and scalar multiplication follow directly from this definition because we perform those operations coordinate by coordinate. We’...