SOME TYPES OF PROOFS YOU SHOULD BE ABLE TO WRITE

· A function is one-to-one

· A function is onto

· A relation is an equivalence relation

· Prove various algebraic vector space properties using v.s. axioms and definitions

(I’ve referred to these as “picky proofs”)

· A binary operation is associative

· A binary operation is commutative

· That an element is the identity element

· That every element has an inverse

· That a given structure is (or is not) a vector space

· A closure argument ( S is closed under a binary operation )

· Prove that a subset of a vector space is (or is not) a subspace of the vector space

· Prove that a subset of a vector space is a line or a plane in that vector space

· Prove that a given vector is (or is not) a linear combination of a given set of vectors

· Prove that a given set of vectors spans (or does not span) a given space

· Prove that a given set of vectors is linearly independent (or dependent)

· Prove that a given set of vectors is (or is not) a basis

· Prove propositions similar to the Expansion and Contraction Lemmas

· Proofs related to inner products and norms

· Proofs involving linear transformations

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