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 \(a_j v_j\) from both sides
of \(a_1v_1+...+a_mv_m=0\)
- Everything past \(v_j\) has to equal 0.
- So we get \(a_j v_j = -a_1 v_1 - ... - a_{j-1} v_{j-1}\)
- Divide by \(a_j\) and we get 2.22
- Thus, \(v_j\) is a linear combination of the other vectors
- And in the \(span(v_1,...,v_j-1)\)
- What \(v_j\) is it replacing? #question
- It's replacing what's in the "…", which is unclear.. is \(v_j\)
actually in the equation then? Or just in the value? #question
- Now, we can remove the \(j^{th}\) finally, and represent it as the
linear combination of the previous elements
- \(\therefore\) 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.