Help with a proof involving the span of a subset

[paulrb]

Homework Statement

Let S be a subset of a vector space V, and let v be an element of V. Show that span(S) = span(S U {v}) if and only if v is an element of span(S)

Homework Equations

The Attempt at a Solution

I'm honestly not sure how to get started, I've spent time looking through the text but it hasn't helped.

 

[matt grime]

You have to do two things.

1. Suppose that v is in span(S), show that span(S)=span(S u {v}).

2. Show that if span(S u {v}) = span(S) then v is in span(S).

Let's look at 2 and break down what you know.

Definition: span(X) is the space of all linear combinations of objects in X.

You need to show v is in span(S).

You know that v is in span(S u {v}) as it is a linear combination of elements of S U {v} trivially.

You know that span(S u {v}) = span(S).

Hence...

Now try 1.

 

[paulrb]

Thanks for your help, it allowed me to finish the rest of the proof.