Linear Algebra Proof: Linear Independence of Set {v1,...,vn,w}

[gavin1989]

Let S = {v1,…vn} be a linearly independent set in a vector space V. Suppose w is in V, but w is not in span(S). Prove that the set T = {v1,…vn,w} is a linearly independent set.



since it says w is in v, does it mean that w is a subspace of V? yet w is not in span(S). I am kinda confused with what it means?


ANy ideas or thoughts on that?

Thanks in advance

 

[matt grime]

You should make sure you honour capitalizations of words. The lower case w is a vector, V is a vector space. The symbols v1 to vn represent n linearly independent vectors. They span a subspace of V.

What would it mean if {v1,..,vn,w} were linearly dependent?

 

[gavin1989]

if {v1,..,vn,w} were linearly dependent, it would span V too.

 

[matt grime]

You have not been told anything spans V. In fact S cannot span V since you are explicitly told that there is a vector, w, that is in V, and not in the span of S.

If {v1,..,vn,w} is a linearly dependent set then <INSERT DEFINITION OF LINEARLY DEPENDENT HERE>.