Associativity of direct sums

Question

Given three vector spaces $U, V$, and $W$, which aren't necessarily subspaces of a common vector space, I have to prove that $(U \oplus V) \oplus W \cong U \oplus (V \oplus W)$. I don't even know how I would begin to approach this, mostly because this is the first time I've encountered direct sums in linear algebra and I'm very fuzzy as to how they actually work.

Answer 1

Well, one could go about this a few ways. The universal property of coproducts will give you associativity. Since this is the first time you've seen it you are likely not familiar with the categorical properties so you could try writing a map from one space to the other. So, where would you send an element

$$((u+v)+w) \in ((U \oplus V) \oplus W)$$

and can you show this is an isomorphism?

Even more, you could also consider the direct product if you find that easier since, in the finite case, the direct sum and direct product are isomorphic. See if that helps.

Direct Sums

As far as how to think about direct sums, they are what we use to decompose algebraic objects in a way that retains linear structure "naturally". This is likely most intuitive in the case of vector spaces. Basic facts from linear algebra will tell you that we can write a vector space in terms of its 1-d subspaces (which trivially) intersect. So, in the case of a 3 dimensional vector space (over $\mathbb{R}$ lets say) we can find a set of linearly independent basis vectors $\{v_1,v_2,v_3\}$ and we have

$$V=\mathbb{R}v_1 \oplus \mathbb{R} v_2 \oplus \mathbb{R} v_3$$

So what does this tell us? Well, for one each of these inherit an $\mathbb{R}$ linear structure from $V$, but even more than that. Every vector in $V$ can be written as a UNIQUE sum of elements from the spaces on the right. This leads to all sorts of nice properties and inheritance of properties for subspaces and so on.

In the case of noninternal structure (if you'll pardon odd phrasing), then direct sum of two arbitrary possibly unrelated vectors spaces $V \oplus W$ we have a formal sum $v+w$ which isn't as "meaningful" in some senses but still quite helpful. In these cases the direct sum and the direct product seem more indistinguishable.

Here are a couple of posts worth reading:

Direct Sum vs. Direct Product vs. Tensor Product