Linear Dependence and Non-Zero Coefficients

[Euler2718]

Homework Statement

True or False:

If $u$, $v$, and $w$ are linearly dependent, then $au+bv+cw=0$ implies at least one of the coefficients $a$, $b$, $c$ is not zero

Homework Equations

Definition of Linear Dependence:

Vectors are linearly dependent if they are not linearly independent; that is there is an equation of the form $c_{1}v_{1}+c_{2}v_{2}+\dots+c_{n}v_{n}$ with at least one coefficient not zero

The Attempt at a Solution

I said true, but the book says false. It gives the reason, "for any vectors $u$, $v$, $w$ - linearly dependent or not - $0u+0v+0w = 0$" . But isn't the problem a direct restatement of the definition? Or am I missing something subtle here.

 

[PeroK]

Homework Statement

True or False:

If $u$, $v$, and $w$ are linearly dependent, then $au+bv+cw=0$ implies at least one of the coefficients $a$, $b$, $c$ is not zero

Homework Equations

Definition of Linear Dependence:

Vectors are linearly dependent if they are not linearly independent; that is there is an equation of the form $c_{1}v_{1}+c_{2}v_{2}+\dots+c_{n}v_{n}$ with at least one coefficient not zero

The Attempt at a Solution

I said true, but the book says false. It gives the reason, "for any vectors $u$, $v$, $w$ - linearly dependent or not - $0u+0v+0w = 0$" . But isn't the problem a direct restatement of the definition? Or am I missing something subtle here.

You're missing something subtle.

 

[haruspex]

If u, v, w linearly independent, au+bv+cw=0 implies a=b=c=0.
Inverting that, if u, v, w linearly dependent, au+bv+cw=0 does not imply a=b=c=0. But they still could be 0.

 

[Euler2718]

Thanks