Proving Linear Independence: If v\notin\left\langlev1,...,vk\right\rangle

[annoymage]

Homework Statement

If v1₁,...,v_k,v are linear independent, prove that v\notin

\left\langlev₁,...,v_k\right\rangle

Homework Equations

n/a

The Attempt at a Solution

i can prove it by contrapositive, but I'm curious how to proof it with

"If v1₁,...,v_k,v are linear independent" in the beginning,

any idea? T_T

 

[HallsofIvy]

I really cannot read what you have here.

Are you trying to show that "if \{v_1, v_2, \cdot\cdot\cdot, v_k, v\} is linearly independent, then v is not in the span of \{v_1, v_2, \cdot\cdot\cdot, v_k\}&quot;?<br /> <br /> Since proof by contradiction works nicely, why look for a &quot;direct&quot; proof?

 

[annoymage]

i'm just curious, maybe there is a way that i don't know,

anyway, you wrote

"<br /> \{v_1, v_2, \cdot\cdot\cdot, v_k, v\}<br /> is linearly independent"

is it the same thing as "<br /> v_1, v_2, \cdot\cdot\cdot, v_k, v<br /> are linear independent" ??