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## Homework Statement

Proof 1:

Show that S= {

**v**

_{1},

**v**

_{2}, ...

**v**

_{p}} is a linearly independent set iff A

**x**=

**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

**c**

_{1}

**v**

_{1}+

**c**

_{2}

**v**

_{2}+ ... +

**c**

_{p}

**v**

_{p}= 0, or similarly only a trivial solution exists to the equation A

**x**=

**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 {

**v**

_{1},

**v**

_{2}, ...

**v**

_{p}}, p>=2 is a linearly independent set, then

**v**

_{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 that

**v**

_{p}}, p>=2 is linearly independent, and let

**v**

_{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 exists

**c**

_{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 equation

**c**

_{1}

**v**

_{1}+

**c**

_{2}

**v**

_{2}+ ... +

**c**

_{p}

**v**

_{p}= 0, because

**c**

_{p}must equal -1.

This is a contradiction. {

**v**

_{1},

**v**

_{2}, ...

**v**

_{p}}, p>=2 can't be a linearly independent set while

**v**

_{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, then

**v**

_{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.