Question

e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a vector space under theNeed to use all axioms to prove this is a vector space.

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Answer #1

er ebelat8)= is the definition of addition and kan ar is scalar multiplication

The zero element is such that er ebre, Va E R

So (q+D) e, Va E Rb 0 so zero element is Or

er ebr Vearebrela+0) E V as a, bE Ra bER

brer as a +b= b+a ebr

erebr)e er(ebr ecr=ela+b+c)a as addition is associative

Zero vector we already found out as Or

kar e ka E R ar E V as a, k ER

as e e-a) er1 rhas additive inverse e

k\cdot (e^{ax}+e^{bx})=k\cdot e^{(a+b)x}=e^{k(a+b)x} which equals k\cdot e^{ax}+k\cdot e^{bx}=e^{kax}+e^{kbx}=e^{(ka+kb)x}

(k_1+k_2)e^{ax}=e^{(k_1+k_2)ax}=k_1e^{ax}+k_2e^{ax}

kk2)e= k1(kzer)= eik2aa rDylya esi

1e^{ax}=e^{1\cdot ax}=e^{ax}

All vector space axioms check out. So it is indeed a full fledged vector space!

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