Linearly Independent Vectors and Their Span: The Truth Behind <u,v,w> and <u,v>

[macca1994]

Homework Statement

Suppose u,v,w are linearly independent, is it true that <u,v,w> does not equal <u,v>

Homework Equations

The Attempt at a Solution

I started by defining what it meant to be linearly independent but am unsure where to go from there. I think the statement is true since the span <u,v> won't include any multiple of w but i can't give a solid proof

 

[micromass]

Assume by contradiction that <u,v,w>=<u,v>. Then w\in &lt;u,v&gt;. Thus...

 

[macca1994]

oh i think i get it, by assuming that we can say by definition

w=λ₁u + λ₂v
then there is now a non trivial solution to
λ₁u + λ₂v + λ₃w=0 which contradicts the statement that u,v,w are linearly independent, is that right?

 

[micromass]

That's right!

 

[macca1994]

Sweet, cheers