2 Linear Algebra Proofs about Linear Independence

[alec_tronn]

Homework Statement

Proof 1:
Show that S= {v₁, v₂, ... v_p} is a linearly independent set iff Ax = 0 has only the trivial solution, where the columns of A are composed of the vectors in S. Be sure to state the relationship of the vector x to the vectors in S

2. The attempt at a solution
As far as I can tell (from my book and class), the actual definition of linear independence is
"A set of vectors is linearly independent iff there exists only the trivial solution to c₁v₁ + c₂v₂+ ... + c_pv_p = 0, or similarly only a trivial solution exists to the equation Ax = 0 when the columns of A are composed of the vectors in S. "

I know many implications of this property, but not the steps in between the term and the definition. Is there a more basic definition that I need to start this proof out with? Are the two options I gave in my definition of linear independence really different enough to merit in-between steps?

Homework Statement

Proof 2:
Prove or disprove: If {v₁, v₂, ... v_p}, p>=2 is a linearly independent set, then v_p is not an element of Span {v₁, v₂, ... v_p-1}. 2. The attempt at a solution

I just took a guess at this one:

Assume that v_p}, p>=2 is linearly independent, and let v_p be an element of Span {v₁, v₂, ... v_p-1}.

By definition of spanning, v_p is a linear combination of {v₁, v₂, ... v_p-1} .

By definition of linear combination, there exists c₁v₁ + c₂v₂+ ... + c_pv_p-1 = v_p.

Algebra give us:
c₁v₁ + c₂v₂+ ... + c_pv_p-1 - v_p = 0.

This gives a solution other than the trivial solution (where {c₁, c₂, ... c_p} are all zero) to the equation c₁v₁ + c₂v₂+ ... + c_pv_p = 0, because c_p must equal -1.

This is a contradiction. {v₁, v₂, ... v_p}, p>=2 can't be a linearly independent set while v_p is an element of Span {v₁, v₂, ... v_p-1}. , so if {v₁, v₂, ... v_p}, p>=2 is a linearly independent set, then v_p is not an element of Span {v₁, v₂, ... v_p-1}, as desired. Q.E.D.Any help or hints on either of these would be greatly appreciated. The first one I just need a starting point, and the second one I need to know if I made any mistakes, or if that is a valid proof. Thanks a lot.

 

[EnumaElish]

I agree with the 2nd proof. In the first problem, the statement and the predicate (the definition) seem to be identical -- I am not clear on what is being asked.

 

[radou]

Regarding proof 1, do you know anything about the dimension of the vector space of solutions to a homogenous system? How does it relate to the rank of the system matrix A? What *is* the rank of your matrix A, since it consists of p linear independent vectors?