Math 251 - Alexiades Review for Quiz 2
on §4.1-4.6 Concepts

vector space, subspace

linear combinations, span

linear independence

basis and dimension

Precise definition of

vector space axioms

subspace of a vector space V:   a subset which is itself a v.s. with same operations as V.

linear combination of {v₁,v₂,...,v_k}⊂V :   a vector of the form c₁v₁+c₂v₂+...+c_kv_k with c_i scalars

span of S={v₁,v₂,...,v_k} :   span(S) = {c₁v₁+...+c_kv_k : c_i scalars} = the set of all linear combinations of v_i∈S

linearly independent set {v₁,v₂,...,v_k} in V :  c₁v₁+c₂v₂+...+c_kv_k=0 implies c₁,c₂,...,c_k are all 0

basis of V:   a subset of V which is linearly independent and spans V.

dimension of V:   number of elements in any basis of V :  min # of lin.independent vectors that span V.

Methods: Know How To

decide if a set/subset (with given operations) constitutes a vector space/subspace

decide if a set {v₁,v₂,...,v_k} is linearly independent

express a vector as a linear combination of {v₁,v₂,...,v_k}

decide if a set {v₁,v₂,...,v_k} spans a space/subspace

find a basis (and dimension) for a space/subspace

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Last Updated:   17 Oct 2011