MATH 218 Exam 2 With Solution, Spring 2016

MATH 218: Elementary Linear Algebra with Applications Spring 2015-2016, Quiz 2, Duration: 60 min. = Sas Exercise | Points | Scores 1 iN 2 20 3 15 4 15 5 23 6 10 Total 100 INSTRUCTIONS: (a) Explain your answers in detail and clearly to ensure full credit. (b) No book. No notes. No calculator. 2 Exercise 1. Prove or disprove using an explicit counterexample that the following sets are sub- spaces (either the set is a subspace and you have to prove it or the set is not a subspace and you have to provide a counterexample): i ihn cae — (), = -£(3)% Dh te. Keprs =a “ole A xeq- 2-0. , We ne bea ley Hut U= 1(3)2° x Ab yp~- (b) © points) U, = {P € Ri[X] | P"(0) > o}. N> An @ x ¢ OC) bt Xx Os. (c) (6 points) U; = {A= (: a & 4 € Ns ee BAe Mo Awa (| | Je Os bf | 2(13)-(21) 80s (22-246, o 3 Exercise 2. Let A be a square n x n matrix. ( oO (1) ( points) Recall the definition of the kernel KerA of A. ke fhe TxeR PAX=O7 (2) G points) Prove that if X € KerA then X € KerA?, . =O frat hes chek AX - AAX- A (AX) (3) (3 points) Find a matrix A such that dim KerA = dim KerA?. A - L-( ] An. A A. | (4) (3 points) Find a matrix A such that dim KerA < dim KerA?. o —-—> 7." 4 _ O- a O Ww al = : : a , aa (n-1 aoe pds ~ 0 ——O kA =a A (ms = Free nan abt ) Th wale p vel ‘ “| Hy gil. (5) (2 points) Find a matrix A such th iin al Dave At. (4) Ta A Li Medle oe tee ho Ad ke At (otlers’e de wold bo Hy Jae), (6) (4 points) Is there a matrix A such that 0 = dim KerA < dim KerA? = 12 No. i Ks ( (= A ‘over (Uf om KH Daven fA ow A RAZ O ¢ cath AL fee a Exercise 3. (15 points) Consider the following matrix ( 3) fi -1 02 1 A={ 0 1 -10 -1 ]. -1 0 102 _ ae) f-) oLtlo foi 9 3(3)—s MO }-! oll o - Of O% O-; 12310 fI7-1 O21. 7 ah, O [iM-| O-) | oO [s 0 Oj/gf2 10 Kr Xa, Ka Lde , X3=h, xs =6 fre =) AG = =! XxX = Kab a . L X= Xz ~ 2x, ~X% =As+b+2€ - é Nae ee Run ef Ke Aig re Exercise 4. (15 points) Determine van{(1 2 De iv; a depending on the value of a a bas EGGS ey deen Ne _ sass tod 4, iseeo ae Ny. ahgeath — all, utd = \ Ay a Ay + ht =O | { oO | fi; © -\ “| & © eS rt O A -| O als ne Ay O) ; Teeny | , wae ks STPaya oles 4) So .) & A a-) oO} O a ey ah = b yotudtle Oo oO ( | @! O ——» Oa Oo;0 aoaorao\e C) CO Ga | O Exer cise 5. Consider the set U = a € R3[X] | P(0) = P’(0)}. (a) (7 points) Prove that i? subspace of R3[X]. e rary Yong Ole) = O7o). GE de@, bt Qe Oi. Phie Cop OQ / Z Q n)=Qa) ba i oy U whips BR, DD.