Proving Linear Dependence in Spanned Vectors

[XcKyle93]

Homework Statement

This should be an easy one, I'm just making sure that I'm not screwing up horribly.

Prove that if v is in span(v₁,v₂, ..., v_N), then v, v₁, v₂, ..., v_N are linearly dependent.

Homework Equations

span(v₁,v₂, ..., v_N) = {Ʃa_iv_i}.

The Attempt at a Solution

If v is in span(v₁,v₂, ..., v_N), then for some scalars a₁, ..., a_N, v = Ʃa_iv_i. This means that 0 = Ʃa_iv_i - v, which means that there exists a set of scalars, not all 0, that satisfy the homogeneous equation. This is because the scalar coefficient of v is -1.

 

[LCKurtz]

Homework Statement

This should be an easy one, I'm just making sure that I'm not screwing up horribly.

Prove that if v is in span(v₁,v₂, ..., v_N), then v, v₁, v₂, ..., v_N are linearly independent.

Homework Equations

span(v₁,v₂, ..., v_N) = {Ʃa_iv_i}.

The Attempt at a Solution

.
If v is in span(v₁,v₂, ..., v_N), then for some scalars a₁, ..., a_N, v = Ʃa_iv_i. This means that 0 = Ʃa_iv_i - v, which means that there exists a set of scalars, not all 0, that satisfy the homogeneous equation. This is because the scalar coefficient of v is -1.

You mean dependent of course. But your argument is OK.

 

[XcKyle93]

Okay, thanks. I was just making sure!