(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Proof 1:

Show that S= {v_{1},v_{2}, ...v_{p}} is a linearly independent set iff Ax=0has only the trivial solution, where the columns of A are composed of the vectors in S. Be sure to state the relationship of the vectorxto 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 toc_{1}v_{1}+c_{2}v_{2}+ ... +c_{p}v_{p}= 0, or similarly only a trivial solution exists to the equation Ax=0when 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?

1. The problem statement, all variables and given/known data

Proof 2:

Prove or disprove: If {v_{1},v_{2}, ...v_{p}}, p>=2 is a linearly independent set, thenv_{p}is not an element of Span {v_{1},v_{2}, ...v_{p-1}}.

2. The attempt at a solution

I just took a guess at this one:

Assume thatv_{p}}, p>=2 is linearly independent, and letv_{p}be an element of Span {v_{1},v_{2}, ...v_{p-1}}.

By definition of spanning,v_{p}is a linear combination of {v_{1},v_{2}, ...v_{p-1}} .

By definition of linear combination, there existsc_{1}v_{1}+c_{2}v_{2}+ ... +c_{p}v_{p-1}=v_{p}.

Algebra give us:

c_{1}v_{1}+c_{2}v_{2}+ ... +c_{p}v_{p-1}-v_{p}= 0.

This gives a solution other than the trivial solution (where {c_{1},c_{2}, ...c_{p}} are all zero) to the equationc_{1}v_{1}+c_{2}v_{2}+ ... +c_{p}v_{p}= 0, becausec_{p}must equal -1.

This is a contradiction. {v_{1},v_{2}, ...v_{p}}, p>=2 can't be a linearly independent set whilev_{p}is an element of Span {v_{1},v_{2}, ...v_{p-1}}. , so if {v_{1},v_{2}, ...v_{p}}, p>=2 is a linearly independent set, thenv_{p}is not an element of Span {v_{1},v_{2}, ...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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: 2 Linear Algebra Proofs about Linear Independence

**Physics Forums | Science Articles, Homework Help, Discussion**