Question

Mark the answers as ”TRUE” or "FALSE” on the front sheet. 1. Let f: Rd + R be continuous. Then the set {x e Rd : f(x) = 5} is closed. 2. Let f: Rd → R be continuous. Then the set {x € Rd: f(x) < 5} is open. 3. The rank of a p xq matrix is equal to min(p, q). 4. There exists a one-to-one linear map L : R5 → R4 5. There exists a surjective linear map L: R4 → R5

For intro to analysis. Please answer true or false and give justification for your answer. Thanks!

Answer 1

Lo fI Rd R be continous then the set 1 s=dn E Rafla)=5} is closed two Auf set s is the set in verse image of $53: Also {53, singleton get is closed in R so Under continoy fune pet og innese image og closed set is also closed set, ie s= Snerti xuftsy 12 f I Rd R be continocy. Then the set s = lnc- Rod! fln) 25} is open (X) Auf f(n) is continary and flo) 25 e s= {xGrdi f(a) € (-005] } but Coos] is closed set in R to inverse image ay (-00 SJ ie f 16.00 5] =s is also closed. Also countor Ex f'RAR be contin sot fln)an then s=dner i foss} = (2005 closed ③ Rankey a PXq matrix < min (p, q) EX AC Ooo here ps 2, 923 but rank A=0 72 so Que (3) also (x) 4 There exist a one to one linear nop LR ART Ang since the null space is the subspace X) a domain space which is R5 and Range space is the subace of codomain space re which is Rt. i NCL) 24

but n(L) + F(L) = 5 [lim ey RE = 8 .. n(L) = 5-86L) → n (L)> 1 [ ELL) <4 ] Livyw is one one if n(L) = 0 but here n(L)>, so 7 any one one map LiRS AR A o 7 a subjective linear map LIR?RS (x r(L) + n(L) = dim of RA (R) &LL) +n(L) = 4 7 &LL) <4 & 5 - Range space L & RS - not onto mapping exist. Ī