A Question about Linear Dependence

[The Student]

My notes have the following in bold.

Proposition. If X is linearly dependent, and X is a subset of a finite set Y , then Y is also linearly dependent.
Proof. Suppose that
X = {v1,...,vm}, Y = {v1,...,vm,w1,...,wk}.
Since X is linearly dependent, then there are scalars a1,...,am, not all 0, such that a1v1 + ··· + am, vm = 0.
Then
a1v1 + ··· + am, vm + 0w1 + ··· + 0wm = 0
so Y is linearly dependent.

The proposition seems to be false. If the coefficients for the set of the w vectors are not zero, and the w vectors are not scalar combinations of v vectors, then wouldn't that mean that some of Y is linearly independent to X?

 

[pka]

Proposition. If X is linearly dependent, and X is a subset of a finite set Y , then Y is also linearly dependent. Proof. Suppose that
X = {v1,...,vm}, Y = {v1,...,vm,w1,...,wk}.
Since X is linearly dependent, then there are scalars a1,...,am, not all 0, such that a1v1 + ··· + am, vm = 0.
Then
a1v1 + ··· + am, vm + 0w1 + ··· + 0wm = 0
so Y is linearly dependent.

The proposition seems to be false. If the coefficients for the set of the w vectors are not zero, and the w vectors are not scalar combinations of v vectors, then wouldn't that mean that some of Y is linearly independent to X?

Consider this:
\(\displaystyle X = \left\{ {\left( {1,0,0)} \right),\left( {0,1,0} \right)} \right\}\,\& \,Y = \left\{ {\left( {1,0,0)} \right),\left( {0,1,0} \right),\left( {1,1,0)} \right)} \right\}\,\)

 

[The Student]

Consider this:
\(\displaystyle X = \left\{ {\left( {1,0,0)} \right),\left( {0,1,0} \right)} \right\}\,\& \,Y = \left\{ {\left( {1,0,0)} \right),\left( {0,1,0} \right),\left( {1,1,0)} \right)} \right\}\,\)

Yeah thanks, it just clicked.