I understand how direct sums work and how they can be useful in proving certain conditional statements in linear algebra but it seems to me that direct sums are only useful in abstract settings. I was wondering if there are any real world applications for direct sums and if so, is there anywhere where I can read more about those applications since I was having trouble finding any.

  • $\begingroup$ It would be impossible to construct any simple example of 2D or higher dimensional vector space without direct sum. Is this a good enough (purely mathematical) reason to consider direct sum? $\endgroup$ – edm Mar 9 at 6:57
  • $\begingroup$ hmmm... I didn't consider that. So then creating a 4d puzzle game like Miegakure (miegakure.com/) would be impossible without use of direct sums. That's cool! $\endgroup$ – Gabriel Chavez Mar 9 at 7:11
  • $\begingroup$ I mean, $\Bbb R^2$ is constructed as $\Bbb R\oplus\Bbb R$, $\Bbb R^3$ is constructed as $\Bbb R\oplus\Bbb R\oplus\Bbb R$, and so on. It is not like higher dimensions are not possible, but is that we could not demonstrate their existence if we restrict the tools we use. And by the tools, I mean direct sum. $\endgroup$ – edm Mar 9 at 7:15
  • $\begingroup$ I see. thanks for the clarification. $\endgroup$ – Gabriel Chavez Mar 9 at 7:18
  • $\begingroup$ How about block matrices? A 2x2 block matrix represents an operator $A: V_1 \oplus V_2 \to W_1 \oplus W_2$, with each block representing some component $A_{ij}: V_j \to W_i$. Any time you see a block matrix, there is a direct sum behind the scenes. $\endgroup$ – Joppy Mar 9 at 7:25

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