Based on the wise words of a crab, I will start writing down some Proof Design Patterns I saw over Axler.
inheriting properties (splitting, doing, merging) “complex numbers inherit commutativity via real numbers”
construct then generalize for uniqueness and existence
zero is cool, and here too!, also \(1-1=0\)
- \(0v = 0\)
- \(1-1 = 0\)
- \(v-v=0\) a.k.a. \(v+(-v)=0\)
- \(v+0 = v\)
distributivity is epic: it is essentially the only tool to connect scalar multiplication and addition in a vector space
“smallest” double containement proofs to show set equivalence: prove one way, then prove the converse (\(a \subset b, b\subset a \Rightarrow a=b\))
couple hints
- step 1: identify
- hypothesis (assumptions)
- desired conclusion (results, trying/to/proof)
- step 2: define
- write down precise, mathematical notations
- step 1: identify
proving uniqueness: set up two distinct results, show that they are the same
proving negation: if the “negative” is distinct, but the direct case is more nebulous, use proves by contradiction
- especially if you are dealing with polynomials, try factoring
- tools to help includes length of linearly-independent list \(\leq\) length of spanning list
Uniqueness by construction: uniqueness part of basis of domain
- pick one element that does exist
- pick arbitrary elements and construct a result
if we are trying to prove equivalence, double-containment is a good bet
see fundamental theorem of linear maps: but basically wehnever you need to construct basis of things start with an arbiturary basis of the subspace and expand into that of the whole space