Processing math: 83%
1 Linear Dependence Lemma
- Why do we care that j is the largest element? #question
- So we can add up everything before it? Just arbitrary?
- Oh, so we can cancel everything after it.
- Can also choose the smallest, it's just about segmenting
- How does 2.22 work? #question
- To get to 2.22, subtract everything but ajvj from both sides
of a1v1+...+amvm=0
- Everything past vj has to equal 0.
- So we get ajvj=−a1v1−...−aj−1vj−1
- Divide by aj and we get 2.22
- Thus, vj is a linear combination of the other vectors
- And in the span(v1,...,vj−1)
- What vj is it replacing? #question
- It's replacing what's in the "…", which is unclear.. is vj
actually in the equation then? Or just in the value? #question
- Now, we can remove the jth finally, and represent it as the
linear combination of the previous elements
- ∴ any element of the span can be represented without
v_j This is called a direct proof! Also, we can iterate this
process until we get a linearly independent list.