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$$\notin$$

$$\left\langle$$v₁,...,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 [itex]\{v_1, v_2, \cdot\cdot\cdot, v_k\}"?

Since proof by contradiction works nicely, why look for a "direct" proof?

 

[annoymage]

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

anyway, you wrote

"$\{v_1, v_2, \cdot\cdot\cdot, v_k, v\}$ is linearly independent"

is it the same thing as "$v_1, v_2, \cdot\cdot\cdot, v_k, v$ are linear independent" ??