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Linear Dependence Definition

1. Homework Statement

True or False:

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

2. 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 [itex]c_{1}v_{1}+c_{2}v_{2}+\dots+c_{n}v_{n}[/itex] with at least one coefficient not zero

3. The Attempt at a Solution

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

PeroK

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1. Homework Statement

True or False:

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

2. 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 [itex]c_{1}v_{1}+c_{2}v_{2}+\dots+c_{n}v_{n}[/itex] with at least one coefficient not zero

3. The Attempt at a Solution

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

haruspex

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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.
 
Thanks
 

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