Question

Need to use all axioms to prove this is a vector space.

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 these definitions? If so, what is the 0 element = eaeba- 8. Let V = k ea of V?

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

as addition is associative

erebr)e er(ebr ecr=ela+b+c)a

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 e'si

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

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