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1. Dec 6, 2014

### mafagafo

1. The problem statement, all variables and given/known data
How to indicate that a vector b is in the span of the columns of a matrix C?

2. Relevant equations
I could type the definition of Span here, but Wikipedia has it too and it is not necessary or useful now.

3. The attempt at a solution
$$\mathbf{b} \in \mathrm{Span}\{\mathbf{c}_1, \mathbf{c}_2, \ldots, \mathbf{c}_n\}$$

I've never seen the $\in$ symbol in this context and wanted to be sure it is OK. As the concept of span seems to be defined formally in such a way that it ends up being a set, I think this operator is the right one.

2. Dec 6, 2014

### Staff: Mentor

The above represents a set of vectors, and $\in$ simply means that b belongs to that set.

3. Dec 6, 2014

### mafagafo

This is a yes, right? (slow guy)

4. Dec 6, 2014

### Staff: Mentor

Yes.

BTW, the span of a set of vectors is a set, albeit an infinite set - the set of all linear combinations of the vectors listed in the set.

5. Dec 6, 2014

### mafagafo

What about $$\mathrm{Span}\{\varnothing\}$$?

6. Dec 7, 2014

### Staff: Mentor

I've never seen this, but I would guess that it's the empty set.

7. Dec 8, 2014

### HallsofIvy

Staff Emeritus
Mark44 should have said "the span of any non-empty set of vectors is infinite".