Indicating Vector Inclusion in Span of Matrix Columns

[mafagafo]

Homework Statement

How to indicate that a vector b is in the span of the columns of a matrix C?

Homework Equations

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

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.

 

[Mark44]

Homework Statement

How to indicate that a vector b is in the span of the columns of a matrix C?

Homework Equations

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

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.

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

 

[mafagafo]

This is a yes, right? (slow guy)

 

[Mark44]

This is a yes, right? (slow guy)

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.

 

[mafagafo]

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.

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

 

[Mark44]

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

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

 

[HallsofIvy]

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