Thread: Prove linearly independent

Originally Posted by Raoh

yes i am.

I think he proving by contradiction, suppose L.D then get L.I

Originally Posted by 450081592

I think he proving by contradiction, suppose L.D then get L.I

he didn't find any contradiction,besides the proof itself needs to be explained.

One way to prove that $\displaystyle P \longrightarrow Q$ is to show that $\displaystyle - Q \longrightarrow -P$, which is what I did. If $\displaystyle u,v,w$ are linearly dependent then $\displaystyle S$ must be linearly dependent. Thus if $\displaystyle S$ is linearly independent we must necessarily have that $\displaystyle u,v,w$ are also linearly independent.

The proof is given. Which part don't you understand?

Originally Posted by Raoh

he didn't find any contradiction,besides the proof itself needs to be explained.

He did find a contradiction- that the set of vectors $\displaystyle S =\{au+bv+cw,\ du+ev+fw,\ gu+hv+iw\}$ was linearly dependent when the hypothesis is that it was independent.

And I thought his proof was very clear.

if $\displaystyle u,v$ and $\displaystyle w$ were dependent,$\displaystyle \text{dim}_{\mathbb{R}}Span(u,v,w)\leq 2$,therefore $\displaystyle u,v$ and $\displaystyle w$ will be in $\displaystyle Span(u,v,w) $ with a dimension less than 2,hence they can't be independent.
is that what you're trying to say
P.S: i like your proof i just can't see through it.