3

As I know span($\emptyset$) $ ={0}$. In other words, the basis of space ${0}$ is empty. Also, I know that any vector in the space is a linear combination of vectors from its basis.

My question is how to express 0 as a linear combination of absent vectors?

MrakarLeo
  • 129

1 Answers1

6

By convention, the empty sum is $$\sum_{x\in\varnothing} x = 0.$$ The intuition for this is that zero is the additive identity. For example, if $S$ is a nonempty finite set of vectors, then as $S\cap\varnothing = \varnothing$, $$\sum_{x\in S\cup\varnothing} x = \sum_{x\in S} x + \sum_{x\in\varnothing} x = \sum_{x\in S} x + 0 = \sum_{x\in S}x.$$ Similarly, the empty product is interpreted as one: $$\prod_{x\in\varnothing} x = 1, $$ Recall that $0!=1$, and that $1$ denotes the multiplicative identity in a field.

Math1000
  • 36,983
  • 1
    I didn't get the "empty sum". You wrote that it's a convention, so I should just accept it and don't think about it, or is there any way to understand why adding nothing produces 0? –  MrakarLeo Jun 08 '15 at 12:03
  • 1
    now I got it, so, essentially, this non-existent sum is forced to be zero! Thank you for explanation. –  MrakarLeo Jun 08 '15 at 12:19