Math 375 - 9/18

Question 1 • Let V be a vector space,S⊆T⊆V be subsets • Prove or disprove: • S independence⇒T independence ○ False ○ Counterexample 1 § V={0} § T={0} § S=∅ ○ Counterexample 2 § V=R2 § T={(0,1),(1,0),(1,1)} § S={(0,1),(1,0)} • T independence⇒S independence ○ True • span(S)=V⇒span(T)=V ○ True • span(T)=V⇒span(S)=V ○ False ○ Counterexample § V=R3 § T={(1,0,0),(0,1,0),(0,0,1)} § S={(1,0,0)} Question 2 • For which functions f:R→R is {f,f^′ } linear dependent • f(x)=〖Ae〗^tx where k≠1 and A≠0