by Seth Grief-Albert

<aside> 💡 Practice proofs. Your proof is always worse than you think it is. That’s my biggest piece of advice.

</aside>

Set Theory

Linear Vector Spaces

Linear Transformations

Basis

Matrix

Eigen

Elementary Proof Strategies

From Numbers, Groups & Codes by Humphreys and Prest

• Argument by contradiction: prove a statement's negation leads to a contradiction.
• Argument by cases: split the range of possibilities into two or more cases and show that all but one leads to a contradiction, so that case must hold.
• Argument by contrapositive: prove $\neg q\to\neg p \text{ rather than } p\to q$
• Choosing the least: when dealing with positive integers, choose the smallest one in the set that satisfies some condition or property. Show that if it did not have this property an even smaller integer could be produced.
• Showing equality indirectly: show that $X\subseteq Y \text{ and } Y\subseteq X$.
• 'Doing the same to both sides': can be used for an equation or inequality
• induction: used to prove "obvious" properties
• Showing that a construction terminates: if at each stage the construction produces, say, a natural number, and these numbers are strictly decreasing at each stage, then by the Well-ordering Principle the construction must stop.
• Use of key results: use major theorems or useful lemmas, such as Lagrange's theorem