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Math 2054 Quiz 6, pagel ~ November 2, 2007 NAME, Sggerted Solshions 1. Suppose b has components 6), b:,63, A is a 3x4 matrix and the augmented matrix corresponding to 1468 by — 2bs the equation Ax = b is row equivalent to | 0 0 1 1) by +43 0 0 0 &| 3b) + 2b, + bg 1A. Suppose k = 1. What conditions (if any) must },,b2 and bg satisfy in order for b to be in Col(A)? Explain! Of A= 1y the system & Conistont. for any & by yee S Hore G00 NO restachens y eee (Wo candihinc) on b, he hs 2 1B. So, if k = 1, is Col(A) all of R3? Explain! G61 GIA=RR? see tr aay 6 ERY Aaa hus a sol, fe 6 es 43 Le f the Gglornns f 1C. Suppose & = 1. Find vectors that span the nullspace of A. Hint: Think about the way we write the solutions of the homogeneous equation Ax = 0 in “parametric form”. i ; . . EA Ged Soe %, 4 = ; [2 ol ® = Ai=0 haw sauturs =| A] [ a af : - he : Senet 8 3 hare x0 free, 6°, ail vectors iy null {) ane wna Piles 4 fs) vector [2] 5 his vector gaans the rullipace. 1D. Suppose k = 0. What conditions (if any) Zr di, by and i satisfy in order for b to be in Col(.A)? Explain! ng, ARR hua sebhm <> 3b,+2b, th =20 (otha se the Syctm)s incomsitent,) Lt in parcola 1E. So, if k =0, is Col(A) all of R37 ceee f rr. = shai th he cobimn Space now, [J el* Le 00h uetdors b= b | lig an en 2A. Is the set H ={f € F | the graph of f passes through the point (0,3)} closed under vector addi- tion? Prove i ive a counterexample. GB. wr clk Br (cmt » Fij=Xt3 at alk) = x3, since lo) =3, LEM, al sace. 9l)= 3, 64% But ous oe 6, i“ te Sroph 4 Fe4 p passer thea. (36) iattead d (2, 3), so Uf Lee Goind fw? specihe abmbea 4 H whose Sam ts NOT in 2B. Is the set G = {f € F | the graph of f passes through the point (3,0)} closed under vector addi- tion? Prove itor give a counterexample. lk fant 9 ie doy aehitong Member gf (Fj ne need ross Fi eG, & #(3)=0 fet 9(3)= 0. Mu, (f+ +9)(2) = 3) +903) =0400, So te alo passes tow (3, ) And ths fog OG, 2. Let F be the vector space of all continuous functions f : R — R, as discussed in class. 2C. Which (if either) of H or G is a subspace of F? G 6 Nor Shee eee i , tl detnte. Quon contnh the © vector! (na DEG Since the creme’ 2 paeraa Pgh (49) 3 ee cae od lis : ‘and tS aaa bh chow HE chased om cmelt, foo 
