Table of Contents
1 Readings
- Axler 2.A
- Under "Linear Independence", what is the whole thing about subtracting equations and "if the only way to do this is the obvious way"? pg.32
- Linear independence feels somewhat okay, but everything past linear dependence lost me.
- Axler 2.C
- Under example 2.41, near the end, why can't \(\text{dim }U\) not
equal 4? Why must you be able to expand it by at least one element?
- Maybe because there are elements in \(\mathcal{P}_m(\mathbb{R})\) that aren't in \(U\), so the basis of \(U\) must be a different length from the basis of \(V\) (else \(U\) would equal \(V\) and all elements of \(V\) would be in \(U\) by 2.39)
- We can shove \(f(x) = x\) into the basis of \(U\) and it will still be linearly independent (because \(f\) was not in \(U\)), so \(\text{dim }U\) must be less than 4.
- Under example 2.41, near the end, why can't \(\text{dim }U\) not
equal 4? Why must you be able to expand it by at least one element?