Linear Transformation

Question

L : V → W is a linear transformation, and v1≠ 0, L(v1) = 0, L(v2) ≠ 0, show that v1, v2 are linearly independent.

Answer 1

I suggest proof by contradiction:

 

Let v1 and v2 be linearly dependent.  Then v2 = a*v1 for some scalar a.

 

Let L(v2) = v3 ≠ 0 (given), and L(v1) = 0 (given)

 

Then

 

L(v1) + L(v2) = L(v1) + L(a*v1) = L(v1) + a*L(v1) = 0 + a*(0) = 0

 

But

 

L(v1) + L(v2) = 0 + v3 = v3 ≠ 0

 

This is a contradiction, so v1 and v2 must be linearly independent.