Linear Algebra
Suppose that S is linearly independent, and that v is a vector in V which is not an element of S. Prove that {v1, v2, . . . , vm, v} is linearly dependent
if and only if 'v' belongs to Span(S).

I have the proof for the this, but wondered if someone could provide a numerical example so that I could better my understanding and apply it to the proof, I would really appreciate it.

Thanks

Question

Linear Algebra
Suppose that S is linearly independent, and that v is a vector in V which is not an element of S. Prove that {v1, v2, . . . , vm, v} is linearly dependent
if and only if 'v' belongs to Span(S).

I have the proof for the this, but wondered if someone could provide a numerical example so that I could better my understanding and apply it to the proof, I would really appreciate it.

Thanks

phi · Answer

I am not sure how a numerical example helps the understanding,
but say you have
[1 0 0] (assume this is a vertical vector)
[0 1 0]

in S

if you add a 3rd vector  that is in the span of S
v=  a [1 0 0] + b [0 1 0]
then the new S is linearly dependent (by definition)

if v = [0 0 1]
it will be independent

anonymous · Answer

Great, thanks so much!
I get it now, I find it quite difficult to work through proofs (I'm pretty new to them) unless I have an example to work through them with and apply to it!)

Thanks again