Proof Techniques

The dimension formula and subset relation

This technique is used when we want to prove two subspace are equal. And it's easy to prove one side is contained in another but it's hard to prove the reverse direction.

Example: 3.130 the range of $𝑇'$

In (a) it first proves $\text{dim range } T = \text{dim range } T'$.

Then in (b), when we try to prove $\text{range } T' = (\text{null } T)^0$, one direction $\text{range } T' \subseteq (\text{null } T)^0$ is easy.

Then we use (a) to show $\text{dim range } T' = \dim (\text{null } T)^0$.

There a couple of exercises in 3F also follow the same approach.

The degree of a polynomial and its zeros