5.10 Direct sums

Our treatment of vector spaces has alternated between the fairly concrete examples of R² and R³ and the more abstract definition presented near the beginning of this chapter. I continue going back and forth with the fundamental idea of the direct sum of two vectors spaces over the same field F.

In a vague and neither concrete not abstract sense, a direct sum is when you push two vector spaces together. It’s one of the ways you can get a new vector space from existing ones.

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

v ⊕ w = (v₁, v₂, …, v_n, w₁, w₂, …, w_m)

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

All the necessary requirements regarding addition and scalar multiplication follow directly from this definition because those operations are done coordinate by coordinate...