Math 375 - 11/15

Theorem • Statement ○ If dim⁡V=dim⁡W<∞, then for linear map T:V→W ○ injective ⟺ surjective ⟺ bijective • Proof ○ By Rank-Nullity Theorem § dim⁡W=dim⁡V=dim⁡N(T)+dim⁡Range(T) ○ If T is injective § ⇒dim⁡N(T)=0 § ⇒dim⁡W=dim⁡Range(T) § ⇒ T is surjective § ⇒T is bijective ○ If T is not injective § ⇒dim⁡N(T)>0 § ⇒dim⁡W≠dim⁡Range(T) § ⇒T is not surjective § ⇒T is not bijective Left Inverse and Right Inverse • If both left inverse and right inverse exists • Then they are the same • Suppose ○ f:V→W ○ g,h:W→V ○ gf=id_V (i.e. g is the left inverse of T) ○ fh=id_w (i.e. h is the right inverse of T) • Then ○ g=g(fh=(gf)h=h Injective and Null Space • Proof: T injective⇒N(T)={0} ○ If T is injective ○ then the only one element mapped to 0 is 0 itself. ○ Therefore N(T)={0} • Proof: N(T)={0}⇒T injective ○ If T(x)=T(y), then ○ T(x)−T(y)=T(x−y)=0 ○ So x−y∈N(T) ○ ⇒x=y ○ Therefore T is injective